\left\{ \begin{array} { l } { \frac { x + 2 y } { 3 } - \frac { x - 2 y } { 5 } = 9 } \\ { \frac { x + 2 y } { 5 } + \frac { x - 2 y } { 4 } = - 1 } \end{array} \right.
Solve for x, y
x=-\frac{25}{74}\approx -0.337837838
y = \frac{1255}{148} = 8\frac{71}{148} \approx 8.47972973
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5\left(x+2y\right)-3\left(x-2y\right)=135
Consider the first equation. Multiply both sides of the equation by 15, the least common multiple of 3,5.
5x+10y-3\left(x-2y\right)=135
Use the distributive property to multiply 5 by x+2y.
5x+10y-3x+6y=135
Use the distributive property to multiply -3 by x-2y.
2x+10y+6y=135
Combine 5x and -3x to get 2x.
2x+16y=135
Combine 10y and 6y to get 16y.
4\left(x+2y\right)+5\left(x-2y\right)=-20
Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4.
4x+8y+5\left(x-2y\right)=-20
Use the distributive property to multiply 4 by x+2y.
4x+8y+5x-10y=-20
Use the distributive property to multiply 5 by x-2y.
9x+8y-10y=-20
Combine 4x and 5x to get 9x.
9x-2y=-20
Combine 8y and -10y to get -2y.
2x+16y=135,9x-2y=-20
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.
2x+16y=135
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
2x=-16y+135
Subtract 16y from both sides of the equation.
x=\frac{1}{2}\left(-16y+135\right)
Divide both sides by 2.
x=-8y+\frac{135}{2}
Multiply \frac{1}{2} times -16y+135.
9\left(-8y+\frac{135}{2}\right)-2y=-20
Substitute -8y+\frac{135}{2} for x in the other equation, 9x-2y=-20.
-72y+\frac{1215}{2}-2y=-20
Multiply 9 times -8y+\frac{135}{2}.
-74y+\frac{1215}{2}=-20
Add -72y to -2y.
-74y=-\frac{1255}{2}
Subtract \frac{1215}{2} from both sides of the equation.
y=\frac{1255}{148}
Divide both sides by -74.
x=-8\times \frac{1255}{148}+\frac{135}{2}
Substitute \frac{1255}{148} for y in x=-8y+\frac{135}{2}. Because the resulting equation contains only one variable, you can solve for x directly.
x=-\frac{2510}{37}+\frac{135}{2}
Multiply -8 times \frac{1255}{148}.
x=-\frac{25}{74}
Add \frac{135}{2} to -\frac{2510}{37} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=-\frac{25}{74},y=\frac{1255}{148}
The system is now solved.
5\left(x+2y\right)-3\left(x-2y\right)=135
Consider the first equation. Multiply both sides of the equation by 15, the least common multiple of 3,5.
5x+10y-3\left(x-2y\right)=135
Use the distributive property to multiply 5 by x+2y.
5x+10y-3x+6y=135
Use the distributive property to multiply -3 by x-2y.
2x+10y+6y=135
Combine 5x and -3x to get 2x.
2x+16y=135
Combine 10y and 6y to get 16y.
4\left(x+2y\right)+5\left(x-2y\right)=-20
Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4.
4x+8y+5\left(x-2y\right)=-20
Use the distributive property to multiply 4 by x+2y.
4x+8y+5x-10y=-20
Use the distributive property to multiply 5 by x-2y.
9x+8y-10y=-20
Combine 4x and 5x to get 9x.
9x-2y=-20
Combine 8y and -10y to get -2y.
2x+16y=135,9x-2y=-20
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}2&16\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}135\\-20\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}2&16\\9&-2\end{matrix}\right))\left(\begin{matrix}2&16\\9&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&16\\9&-2\end{matrix}\right))\left(\begin{matrix}135\\-20\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&16\\9&-2\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}2&16\\9&-2\end{matrix}\right))\left(\begin{matrix}135\\-20\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}2&16\\9&-2\end{matrix}\right))\left(\begin{matrix}135\\-20\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{2}{2\left(-2\right)-16\times 9}&-\frac{16}{2\left(-2\right)-16\times 9}\\-\frac{9}{2\left(-2\right)-16\times 9}&\frac{2}{2\left(-2\right)-16\times 9}\end{matrix}\right)\left(\begin{matrix}135\\-20\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{1}{74}&\frac{4}{37}\\\frac{9}{148}&-\frac{1}{74}\end{matrix}\right)\left(\begin{matrix}135\\-20\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{74}\times 135+\frac{4}{37}\left(-20\right)\\\frac{9}{148}\times 135-\frac{1}{74}\left(-20\right)\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{25}{74}\\\frac{1255}{148}\end{matrix}\right)
Do the arithmetic.
x=-\frac{25}{74},y=\frac{1255}{148}
Extract the matrix elements x and y.
5\left(x+2y\right)-3\left(x-2y\right)=135
Consider the first equation. Multiply both sides of the equation by 15, the least common multiple of 3,5.
5x+10y-3\left(x-2y\right)=135
Use the distributive property to multiply 5 by x+2y.
5x+10y-3x+6y=135
Use the distributive property to multiply -3 by x-2y.
2x+10y+6y=135
Combine 5x and -3x to get 2x.
2x+16y=135
Combine 10y and 6y to get 16y.
4\left(x+2y\right)+5\left(x-2y\right)=-20
Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4.
4x+8y+5\left(x-2y\right)=-20
Use the distributive property to multiply 4 by x+2y.
4x+8y+5x-10y=-20
Use the distributive property to multiply 5 by x-2y.
9x+8y-10y=-20
Combine 4x and 5x to get 9x.
9x-2y=-20
Combine 8y and -10y to get -2y.
2x+16y=135,9x-2y=-20
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.
9\times 2x+9\times 16y=9\times 135,2\times 9x+2\left(-2\right)y=2\left(-20\right)
To make 2x and 9x equal, multiply all terms on each side of the first equation by 9 and all terms on each side of the second by 2.
18x+144y=1215,18x-4y=-40
Simplify.
18x-18x+144y+4y=1215+40
Subtract 18x-4y=-40 from 18x+144y=1215 by subtracting like terms on each side of the equal sign.
144y+4y=1215+40
Add 18x to -18x. Terms 18x and -18x cancel out, leaving an equation with only one variable that can be solved.
148y=1215+40
Add 144y to 4y.
148y=1255
Add 1215 to 40.
y=\frac{1255}{148}
Divide both sides by 148.
9x-2\times \frac{1255}{148}=-20
Substitute \frac{1255}{148} for y in 9x-2y=-20. Because the resulting equation contains only one variable, you can solve for x directly.
9x-\frac{1255}{74}=-20
Multiply -2 times \frac{1255}{148}.
9x=-\frac{225}{74}
Add \frac{1255}{74} to both sides of the equation.
x=-\frac{25}{74}
Divide both sides by 9.
x=-\frac{25}{74},y=\frac{1255}{148}
The system is now solved.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\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 x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}