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from <a href="/w/index.php?title=Equating_the_coefficients&redirect=no" class="mw-redirect" title="Equating the coefficients">Equating the coefficients</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the method of <b>equating the coefficients</b> is a way of solving a functional equation of two expressions such as <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> for a number of unknown <a href="/wiki/Parameter" title="Parameter">parameters</a>. It relies on the fact that two expressions are identical precisely when corresponding <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> are equal for each different type of term. The method is used to bring <a href="/wiki/Formula" title="Formula">formulas</a> into a desired form. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Example_in_real_fractions">Example in real fractions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equating_coefficients&action=edit&section=1" title="Edit section: Example in real fractions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose we want to apply <a href="/wiki/Partial_fraction_decomposition" title="Partial fraction decomposition">partial fraction decomposition</a> to the expression: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{x(x-1)(x-2)}},\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x(x-1)(x-2)}},\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9e54f6d0660097b70afc8884d63f2c05cb0e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.483ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{x(x-1)(x-2)}},\,}" /></span></dd></dl> <p>that is, we want to bring it into the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A}{x}}+{\frac {B}{x-1}}+{\frac {C}{x-2}},\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A}{x}}+{\frac {B}{x-1}}+{\frac {C}{x-2}},\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be0aa80e56531120f1286e088f2878e40620829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.631ex; height:5.509ex;" alt="{\displaystyle {\frac {A}{x}}+{\frac {B}{x-1}}+{\frac {C}{x-2}},\,}" /></span></dd></dl> <p>in which the unknown parameters are <i>A</i>, <i>B</i> and <i>C</i>. Multiplying these formulas by <i>x</i>(<i>x</i> − 1)(<i>x</i> − 2) turns both into polynomials, which we equate: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x-1)(x-2)+Bx(x-2)+Cx(x-1)=1,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>B</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x-1)(x-2)+Bx(x-2)+Cx(x-1)=1,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a21adb05a25c3794adbfd9296a1b3dcab63ff143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.476ex; height:2.843ex;" alt="{\displaystyle A(x-1)(x-2)+Bx(x-2)+Cx(x-1)=1,\,}" /></span></dd></dl> <p>or, after expansion and collecting terms with equal powers of <i>x</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A+B+C)x^{2}-(3A+2B+C)x+2A=1.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>A</mi> <mo>+</mo> <mn>2</mn> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>A</mi> <mo>=</mo> <mn>1.</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A+B+C)x^{2}-(3A+2B+C)x+2A=1.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9038964f5026c5cf52f03722006dca99fbc678c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.447ex; height:3.176ex;" alt="{\displaystyle (A+B+C)x^{2}-(3A+2B+C)x+2A=1.\,}" /></span></dd></dl> <p>At this point it is essential to realize that the polynomial 1 is in fact equal to the polynomial 0<i>x</i><sup>2</sup> + 0<i>x</i> + 1, having zero coefficients for the positive powers of <i>x</i>. Equating the corresponding coefficients now results in this <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B+C=0,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B+C=0,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36940e9f4f8af40e47d74cf08f33adf6d8237c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.249ex; height:2.509ex;" alt="{\displaystyle A+B+C=0,\,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3A+2B+C=0,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>A</mi> <mo>+</mo> <mn>2</mn> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3A+2B+C=0,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e3fe3cee9c98ea651368a44e23157926ef3df9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.574ex; height:2.509ex;" alt="{\displaystyle 3A+2B+C=0,\,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2A=1.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>A</mi> <mo>=</mo> <mn>1.</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2A=1.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c75ef36382dd7847d589570d40b1e7dd44957c80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.2ex; height:2.176ex;" alt="{\displaystyle 2A=1.\,}" /></span></dd></dl> <p>Solving it results in: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {1}{2}},\,B=-1,\,C={\frac {1}{2}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{2}},\,B=-1,\,C={\frac {1}{2}}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2b403fe43b670e30b53e1537cf5010dcc28c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.413ex; height:5.176ex;" alt="{\displaystyle A={\frac {1}{2}},\,B=-1,\,C={\frac {1}{2}}.\,}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Example_in_nested_radicals">Example in nested radicals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equating_coefficients&action=edit&section=2" title="Edit section: Example in nested radicals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the <a href="/wiki/Nested_radical" title="Nested radical">nested radicals</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mtext> </mtext> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f105f03400b9e104e3975d37b5445aee55317f61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.915ex; height:4.843ex;" alt="{\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}}" /></span> to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of <a href="/wiki/Rational_number" title="Rational number">rational</a> parameters <i>d, e</i> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}={\sqrt {d}}+{\sqrt {e}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mtext> </mtext> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>e</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}={\sqrt {d}}+{\sqrt {e}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5f9c9e962a4b46020568d66a4e2d6786e03941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.672ex; height:4.843ex;" alt="{\displaystyle {\sqrt {a+b{\sqrt {c}}\ }}={\sqrt {d}}+{\sqrt {e}}.}" /></span></dd></dl> <p>Squaring both sides of this equation yields: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b{\sqrt {c}}=d+e+2{\sqrt {de}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mi>d</mi> <mo>+</mo> <mi>e</mi> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <mi>e</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {c}}=d+e+2{\sqrt {de}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0348b1250a9cd71a620ee139d2366569ad28ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.134ex; height:3.343ex;" alt="{\displaystyle a+b{\sqrt {c}}=d+e+2{\sqrt {de}}.}" /></span></dd></dl> <p>To find <i>d</i> and <i>e</i> we equate the terms not involving square roots, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=d+e,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>d</mi> <mo>+</mo> <mi>e</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=d+e,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ff80ae6bc55fa4aed47e085ec225d159401912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.115ex; height:2.509ex;" alt="{\displaystyle a=d+e,}" /></span> and equate the parts involving radicals, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b{\sqrt {c}}=2{\sqrt {de}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <mi>e</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b{\sqrt {c}}=2{\sqrt {de}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/920371293be7f1ee2af495860fb7c087cbf232e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.436ex; height:3.343ex;" alt="{\displaystyle b{\sqrt {c}}=2{\sqrt {de}}}" /></span> which when squared implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2}c=4de.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> <mo>=</mo> <mn>4</mn> <mi>d</mi> <mi>e</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}c=4de.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6433ded052ee9541b63adc29f6a106e672b7b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.266ex; height:2.676ex;" alt="{\displaystyle b^{2}c=4de.}" /></span> This gives us two equations, one <a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic</a> and one linear, in the desired parameters <i>d</i> and <i>e</i>, and these <a href="/wiki/Nested_radical#Denesting" title="Nested radical">can be solved</a> to obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\frac {a+{\sqrt {a^{2}-b^{2}c}}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\frac {a+{\sqrt {a^{2}-b^{2}c}}}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e6f33ccee923eaa12cb4ae49b911437c838c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.242ex; height:6.176ex;" alt="{\displaystyle e={\frac {a+{\sqrt {a^{2}-b^{2}c}}}{2}},}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {a-{\sqrt {a^{2}-b^{2}c}}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\frac {a-{\sqrt {a^{2}-b^{2}c}}}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7348406c2774bcb4d8c764c810d055e78769c757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.374ex; height:6.176ex;" alt="{\displaystyle d={\frac {a-{\sqrt {a^{2}-b^{2}c}}}{2}},}" /></span></dd></dl> <p>which is a valid solution pair <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {a^{2}-b^{2}c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {a^{2}-b^{2}c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d32e93f7dcb928515f491a153fd4e70c1ef91e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.507ex; height:3.509ex;" alt="{\displaystyle {\sqrt {a^{2}-b^{2}c}}}" /></span> is a rational number. </p> <div class="mw-heading mw-heading2"><h2 id="Example_of_testing_for_linear_dependence_of_equations">Example of testing for linear dependence of equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equating_coefficients&action=edit&section=3" title="Edit section: Example of testing for linear dependence of equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider this <a href="/wiki/Overdetermined_system" title="Overdetermined system">overdetermined system of equations</a> (with 3 equations in just 2 unknowns): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-2y+1=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-2y+1=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e708d4fc672bda9e59dc7cb5a9341b7c6df67aed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.399ex; height:2.509ex;" alt="{\displaystyle x-2y+1=0,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+5y-8=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mi>y</mi> <mo>−</mo> <mn>8</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+5y-8=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc83177332c1c530793a394fd45c2c1b5ffd70bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.561ex; height:2.509ex;" alt="{\displaystyle 3x+5y-8=0,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x+3y-7=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo>−</mo> <mn>7</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x+3y-7=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bb0e69a9746464c2175bd73608aeac2cd0228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.561ex; height:2.509ex;" alt="{\displaystyle 4x+3y-7=0.}" /></span></dd></dl> <p>To test whether the third equation is <a href="/wiki/Linear_dependence" class="mw-redirect" title="Linear dependence">linearly dependent</a> on the first two, postulate two parameters <i>a</i> and <i>b</i> such that <i>a</i> times the first equation plus <i>b</i> times the second equation equals the third equation. Since this always holds for the right sides, all of which are 0, we merely need to require it to hold for the left sides as well: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(x-2y+1)+b(3x+5y-8)=4x+3y-7.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mi>y</mi> <mo>−</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo>−</mo> <mn>7.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(x-2y+1)+b(3x+5y-8)=4x+3y-7.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8debd3d3e7c10421ae2cde6a65aa6022bcd6f692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.229ex; height:2.843ex;" alt="{\displaystyle a(x-2y+1)+b(3x+5y-8)=4x+3y-7.}" /></span></dd></dl> <p>Equating the coefficients of <i>x</i> on both sides, equating the coefficients of <i>y</i> on both sides, and equating the constants on both sides gives the following system in the desired parameters <i>a</i>, <i>b</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+3b=4,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>b</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+3b=4,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3ffcff85314d4630bb6f19adbbbc893a774653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.138ex; height:2.509ex;" alt="{\displaystyle a+3b=4,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2a+5b=3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>5</mn> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2a+5b=3,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffab2a597622ca9acef5f78daccdfc9cdb1bed67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.109ex; height:2.509ex;" alt="{\displaystyle -2a+5b=3,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-8b=-7.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mn>8</mn> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>7.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-8b=-7.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13893c60ff19a6a3b5011d848ac8407eb9e1f5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.946ex; height:2.343ex;" alt="{\displaystyle a-8b=-7.}" /></span></dd></dl> <p>Solving it gives: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1,\ b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=1,\ b=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96514b0b98a9b7d7a22b5eb6fc8eb7bd571b76d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.364ex; height:2.509ex;" alt="{\displaystyle a=1,\ b=1}" /></span></dd></dl> <p>The unique pair of values <i>a</i>, <i>b</i> satisfying the first two equations is (<i>a</i>, <i>b</i>) = (1, 1); since these values also satisfy the third equation, there do in fact exist <i>a</i>, <i>b</i> such that <i>a</i> times the original first equation plus <i>b</i> times the original second equation equals the original third equation; we conclude that the third equation is linearly dependent on the first two. </p><p>Note that if the constant term in the original third equation had been anything other than –7, the values (<i>a</i>, <i>b</i>) = (1, 1) that satisfied the first two equations in the parameters would not have satisfied the third one (<i>a</i> – 8<i>b</i> = constant), so there would exist no <i>a</i>, <i>b</i> satisfying all three equations in the parameters, and therefore the third original equation would be independent of the first two. </p> <div class="mw-heading mw-heading2"><h2 id="Example_in_complex_numbers">Example in complex numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equating_coefficients&action=edit&section=4" title="Edit section: Example in complex numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The method of equating coefficients is often used when dealing with <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. For example, to divide the complex number <i>a</i>+<i>bi</i> by the complex number <i>c</i>+<i>di</i>, we postulate that the ratio equals the complex number <i>e+fi</i>, and we wish to find the values of the parameters <i>e</i> and <i>f</i> for which this is true. We write </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a+bi}{c+di}}=e+fi,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mrow> <mrow> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>e</mi> <mo>+</mo> <mi>f</mi> <mi>i</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a+bi}{c+di}}=e+fi,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44ec48ea8398c1b87cc413670028190658956269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.457ex; height:5.676ex;" alt="{\displaystyle {\frac {a+bi}{c+di}}=e+fi,}" /></span></dd></dl> <p>and multiply both sides by the denominator to obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (ce-fd)+(ed+cf)i=a+bi.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mi>e</mi> <mo>−</mo> <mi>f</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mi>d</mi> <mo>+</mo> <mi>c</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mi>i</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ce-fd)+(ed+cf)i=a+bi.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a65e3101b631400e21e54a4c5579861f10548b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.727ex; height:2.843ex;" alt="{\displaystyle (ce-fd)+(ed+cf)i=a+bi.}" /></span></dd></dl> <p>Equating <a href="/wiki/Real_number" title="Real number">real</a> terms gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ce-fd=a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>e</mi> <mo>−</mo> <mi>f</mi> <mi>d</mi> <mo>=</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ce-fd=a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f62c714ec89a3ef906f116126526c5aaf40bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.4ex; height:2.509ex;" alt="{\displaystyle ce-fd=a,}" /></span></dd></dl> <p>and equating coefficients of the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> <i>i</i> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ed+cf=b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi>d</mi> <mo>+</mo> <mi>c</mi> <mi>f</mi> <mo>=</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ed+cf=b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a683975c00a260adf9c6efd5cedc39db8cd9a8d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.168ex; height:2.509ex;" alt="{\displaystyle ed+cf=b.}" /></span></dd></dl> <p>These are two equations in the unknown parameters <i>e</i> and <i>f</i>, and they can be solved to obtain the desired coefficients of the quotient: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\frac {ac+bd}{c^{2}+d^{2}}}\quad \quad {\text{and}}\quad \quad f={\frac {bc-ad}{c^{2}+d^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>c</mi> <mo>+</mo> <mi>b</mi> <mi>d</mi> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="1em"></mspace> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em"></mspace> <mspace width="1em"></mspace> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>c</mi> <mo>−</mo> <mi>a</mi> <mi>d</mi> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\frac {ac+bd}{c^{2}+d^{2}}}\quad \quad {\text{and}}\quad \quad f={\frac {bc-ad}{c^{2}+d^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f861895d3da85e4ad41d823aaefa0f2e715e6cf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.497ex; height:5.843ex;" alt="{\displaystyle e={\frac {ac+bd}{c^{2}+d^{2}}}\quad \quad {\text{and}}\quad \quad f={\frac {bc-ad}{c^{2}+d^{2}}}.}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equating_coefficients&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFTanton2005" class="citation book cs1">Tanton, James (2005). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/encyclopediamath00tant_807"><i>Encyclopedia of Mathematics</i></a></span>. Facts on File. p. <a rel="nofollow" class="external text" href="https://archive.org/details/encyclopediamath00tant_807/page/n170">162</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8160-5124-0" title="Special:BookSources/0-8160-5124-0"><bdi>0-8160-5124-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+Mathematics&rft.pages=162&rft.pub=Facts+on+File&rft.date=2005&rft.isbn=0-8160-5124-0&rft.aulast=Tanton&rft.aufirst=James&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fencyclopediamath00tant_807&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquating+coefficients" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Equating_coefficients&oldid=1164721533">https://en.wikipedia.org/w/index.php?title=Equating_coefficients&oldid=1164721533</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Elementary_algebra" title="Category:Elementary algebra">Elementary algebra</a></li><li><a href="/wiki/Category:Equations" title="Category:Equations">Equations</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 July 2023, at 18:10<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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