Algebra InputsTrigonometry InputsCalculus InputsMatrix Inputs Basic algebra trigonometry calculus statistics matrices Characters \left\{ \begin{array} { l } { \frac { x } { m } + \frac { y } { n } = 2 m } \\ { m x - n y = m ^ { 3 } - m n ^ { 2 } } \end{array} \right.Solve for x, y x=m^{2} y=mn n\neq 0\text{ and }m\neq 0Steps Using SubstitutionSteps Using MatricesSteps Using EliminationView solution stepsSteps Using Substitution \left\{ \begin{array} { l } { \frac { x } { m } + \frac { y } { n } = 2 m } \\ { m x - n y = m ^ { 3 } - m n ^ { 2 } } \end{array} \right. To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation. \frac{1}{m}x+\frac{1}{n}y=2m,mx+\left(-n\right)y=m^{3}-mn^{2} Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign. \frac{1}{m}x+\frac{1}{n}y=2m Subtract \frac{y}{n} from both sides of the equation. \frac{1}{m}x=\left(-\frac{1}{n}\right)y+2m Multiply both sides by m. x=m\left(\left(-\frac{1}{n}\right)y+2m\right) Multiply m times 2m-\frac{y}{n}. x=\left(-\frac{m}{n}\right)y+2m^{2} Substitute \frac{m\left(-y+2mn\right)}{n} for x in the other equation, mx+\left(-n\right)y=m^{3}-mn^{2}. m\left(\left(-\frac{m}{n}\right)y+2m^{2}\right)+\left(-n\right)y=m^{3}-mn^{2} Multiply m times \frac{m\left(-y+2mn\right)}{n}. \left(-\frac{m^{2}}{n}\right)y+2m^{3}+\left(-n\right)y=m^{3}-mn^{2} Add -\frac{m^{2}y}{n} to -ny. \left(-\frac{m^{2}}{n}-n\right)y+2m^{3}=m^{3}-mn^{2} Subtract 2m^{3} from both sides of the equation. \left(-\frac{m^{2}}{n}-n\right)y=-m\left(m^{2}+n^{2}\right) Divide both sides by -\frac{m^{2}}{n}-n. y=mn Substitute mn for y in x=\left(-\frac{m}{n}\right)y+2m^{2}. Because the resulting equation contains only one variable, you can solve for x directly. x=\left(-\frac{m}{n}\right)mn+2m^{2} Multiply -\frac{m}{n} times mn. x=-m^{2}+2m^{2} Add 2m^{2} to -m^{2}. x=m^{2} The system is now solved. x=m^{2},y=mn GraphQuizSimultaneous Equation5 problems similar to: \left\{ \begin{array} { l } { \frac { x } { m } + \frac { y } { n } = 2 m } \\ { m x - n y = m ^ { 3 } - m n ^ { 2 } } \end{array} \right.
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CopyCopied to clipboard\frac{1}{m}x+\frac{1}{n}y=2m,mx+\left(-n\right)y=m^{3}-mn^{2} To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.\frac{1}{m}x+\frac{1}{n}y=2m Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.\frac{1}{m}x=\left(-\frac{1}{n}\right)y+2m Subtract \frac{y}{n} from both sides of the equation.x=m\left(\left(-\frac{1}{n}\right)y+2m\right) Multiply both sides by m.x=\left(-\frac{m}{n}\right)y+2m^{2} Multiply m times 2m-\frac{y}{n}.m\left(\left(-\frac{m}{n}\right)y+2m^{2}\right)+\left(-n\right)y=m^{3}-mn^{2} Substitute \frac{m\left(-y+2mn\right)}{n} for x in the other equation, mx+\left(-n\right)y=m^{3}-mn^{2}.\left(-\frac{m^{2}}{n}\right)y+2m^{3}+\left(-n\right)y=m^{3}-mn^{2} Multiply m times \frac{m\left(-y+2mn\right)}{n}.\left(-\frac{m^{2}}{n}-n\right)y+2m^{3}=m^{3}-mn^{2} Add -\frac{m^{2}y}{n} to -ny.\left(-\frac{m^{2}}{n}-n\right)y=-m\left(m^{2}+n^{2}\right) Subtract 2m^{3} from both sides of the equation.y=mn Divide both sides by -\frac{m^{2}}{n}-n.x=\left(-\frac{m}{n}\right)mn+2m^{2} Substitute mn for y in x=\left(-\frac{m}{n}\right)y+2m^{2}. Because the resulting equation contains only one variable, you can solve for x directly.x=-m^{2}+2m^{2} Multiply -\frac{m}{n} times mn.x=m^{2} Add 2m^{2} to -m^{2}.x=m^{2},y=mn The system is now solved.\frac{1}{m}x+\frac{1}{n}y=2m,mx+\left(-n\right)y=m^{3}-mn^{2} Put the equations in standard form and then use matrices to solve the system of equations.\left(\begin{matrix}\frac{1}{m}&\frac{1}{n}\\m&-n\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2m\\m\left(m-n\right)\left(m+n\right)\end{matrix}\right) Write the equations in matrix form.inverse(\left(\begin{matrix}\frac{1}{m}&\frac{1}{n}\\m&-n\end{matrix}\right))\left(\begin{matrix}\frac{1}{m}&\frac{1}{n}\\m&-n\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{m}&\frac{1}{n}\\m&-n\end{matrix}\right))\left(\begin{matrix}2m\\m\left(m-n\right)\left(m+n\right)\end{matrix}\right) Left multiply the equation by the inverse matrix of \left(\begin{matrix}\frac{1}{m}&\frac{1}{n}\\m&-n\end{matrix}\right).\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{m}&\frac{1}{n}\\m&-n\end{matrix}\right))\left(\begin{matrix}2m\\m\left(m-n\right)\left(m+n\right)\end{matrix}\right) The product of a matrix and its inverse is the identity matrix.\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{m}&\frac{1}{n}\\m&-n\end{matrix}\right))\left(\begin{matrix}2m\\m\left(m-n\right)\left(m+n\right)\end{matrix}\right) Multiply the matrices on the left hand side of the equal sign.\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{n}{\frac{1}{m}\left(-n\right)-\frac{1}{n}m}&-\frac{\frac{1}{n}}{\frac{1}{m}\left(-n\right)-\frac{1}{n}m}\\-\frac{m}{\frac{1}{m}\left(-n\right)-\frac{1}{n}m}&\frac{1}{m\left(\frac{1}{m}\left(-n\right)-\frac{1}{n}m\right)}\end{matrix}\right)\left(\begin{matrix}2m\\m\left(m-n\right)\left(m+n\right)\end{matrix}\right) For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{mn^{2}}{m^{2}+n^{2}}&\frac{m}{m^{2}+n^{2}}\\\frac{nm^{2}}{m^{2}+n^{2}}&-\frac{n}{m^{2}+n^{2}}\end{matrix}\right)\left(\begin{matrix}2m\\m\left(m-n\right)\left(m+n\right)\end{matrix}\right) Do the arithmetic.\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{mn^{2}}{m^{2}+n^{2}}\times 2m+\frac{m}{m^{2}+n^{2}}m\left(m-n\right)\left(m+n\right)\\\frac{nm^{2}}{m^{2}+n^{2}}\times 2m+\left(-\frac{n}{m^{2}+n^{2}}\right)m\left(m-n\right)\left(m+n\right)\end{matrix}\right) Multiply the matrices.\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}m^{2}\\mn\end{matrix}\right) Do the arithmetic.x=m^{2},y=mn Extract the matrix elements x and y.\frac{1}{m}x+\frac{1}{n}y=2m,mx+\left(-n\right)y=m^{3}-mn^{2} In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.m\times \frac{1}{m}x+m\times \frac{1}{n}y=m\times 2m,\frac{1}{m}mx+\frac{1}{m}\left(-n\right)y=\frac{1}{m}\left(m^{3}-mn^{2}\right) To make \frac{x}{m} and mx equal, multiply all terms on each side of the first equation by m and all terms on each side of the second by m^{-1}.x+\frac{m}{n}y=2m^{2},x+\left(-\frac{n}{m}\right)y=\left(m-n\right)\left(m+n\right) Simplify.x-x+\frac{m}{n}y+\frac{n}{m}y=2m^{2}+n^{2}-m^{2} Subtract x+\left(-\frac{n}{m}\right)y=\left(m-n\right)\left(m+n\right) from x+\frac{m}{n}y=2m^{2} by subtracting like terms on each side of the equal sign.\frac{m}{n}y+\frac{n}{m}y=2m^{2}+n^{2}-m^{2} Add x to -x. Terms x and -x cancel out, leaving an equation with only one variable that can be solved.\left(\frac{m}{n}+\frac{n}{m}\right)y=2m^{2}+n^{2}-m^{2} Add \frac{my}{n} to \frac{ny}{m}.\left(\frac{m}{n}+\frac{n}{m}\right)y=m^{2}+n^{2} Add 2m^{2} to -m^{2}+n^{2}.y=mn Divide both sides by \frac{m}{n}+\frac{n}{m}.mx+\left(-n\right)mn=m^{3}-mn^{2} Substitute mn for y in mx+\left(-n\right)y=m^{3}-mn^{2}. Because the resulting equation contains only one variable, you can solve for x directly.mx-mn^{2}=m^{3}-mn^{2} Multiply -n times mn.mx=m^{3} Add mn^{2} to both sides of the equation.x=m^{2} Divide both sides by m.x=m^{2},y=mn The system is now solved.
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Quadratic equation { x } ^ { 2 } - 4 x - 5 = 0Trigonometry 4 \sin \theta \cos \theta = 2 \sin \thetaLinear equation y = 3x + 4Arithmetic 699 * 533Matrix \left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]Simultaneous equation \left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.Differentiation \frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }Integration \int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d xLimits \lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}Back to top
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Steps Using SubstitutionSteps Using MatricesSteps Using EliminationView solution stepsSteps Using Substitution \left\{ \begin{array} { l } { \frac { x } { m } + \frac { y } { n } = 2 m } \\ { m x - n y = m ^ { 3 } - m n ^ { 2 } } \end{array} \right. To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation. \frac{1}{m}x+\frac{1}{n}y=2m,mx+\left(-n\right)y=m^{3}-mn^{2} Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign. \frac{1}{m}x+\frac{1}{n}y=2m Subtract \frac{y}{n} from both sides of the equation. \frac{1}{m}x=\left(-\frac{1}{n}\right)y+2m Multiply both sides by m. x=m\left(\left(-\frac{1}{n}\right)y+2m\right) Multiply m times 2m-\frac{y}{n}. x=\left(-\frac{m}{n}\right)y+2m^{2} Substitute \frac{m\left(-y+2mn\right)}{n} for x in the other equation, mx+\left(-n\right)y=m^{3}-mn^{2}. m\left(\left(-\frac{m}{n}\right)y+2m^{2}\right)+\left(-n\right)y=m^{3}-mn^{2} Multiply m times \frac{m\left(-y+2mn\right)}{n}. \left(-\frac{m^{2}}{n}\right)y+2m^{3}+\left(-n\right)y=m^{3}-mn^{2} Add -\frac{m^{2}y}{n} to -ny. \left(-\frac{m^{2}}{n}-n\right)y+2m^{3}=m^{3}-mn^{2} Subtract 2m^{3} from both sides of the equation. \left(-\frac{m^{2}}{n}-n\right)y=-m\left(m^{2}+n^{2}\right) Divide both sides by -\frac{m^{2}}{n}-n. y=mn Substitute mn for y in x=\left(-\frac{m}{n}\right)y+2m^{2}. Because the resulting equation contains only one variable, you can solve for x directly. x=\left(-\frac{m}{n}\right)mn+2m^{2} Multiply -\frac{m}{n} times mn. x=-m^{2}+2m^{2} Add 2m^{2} to -m^{2}. x=m^{2} The system is now solved. x=m^{2},y=mn GraphQuizSimultaneous Equation5 problems similar to: \left\{ \begin{array} { l } { \frac { x } { m } + \frac { y } { n } = 2 m } \\ { m x - n y = m ^ { 3 } - m n ^ { 2 } } \end{array} \right.
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