Solve {l}{x-y/4+y+2/7=2y-8}{3y+4=8x-3y/3+9y-5x/4}

$$\left\{ \begin{array} { l } { \frac { x - y } { 4 } + \frac { y + 2 } { 7 } = 2 y - 8 } \\ { 3 y + 4 = \frac { 8 x - 3 y } { 3 } + \frac { 9 y - 5 x } { 4 } } \end{array} \right.$$

Solve for x, y

$$x=9$$

$$y=5$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 28, the least common multiple of 4,7.

$$7\left(x-y\right)+4\left(y+2\right)=56y-224$$

Use the distributive property to multiply 7 by x-y.

$$7x-7y+4\left(y+2\right)=56y-224$$

Use the distributive property to multiply 4 by y+2.

$$7x-7y+4y+8=56y-224$$

Combine -7y and 4y to get -3y.

$$7x-3y+8=56y-224$$

Subtract 56y from both sides.

$$7x-3y+8-56y=-224$$

Combine -3y and -56y to get -59y.

$$7x-59y+8=-224$$

Subtract 8 from both sides.

$$7x-59y=-224-8$$

Subtract 8 from -224 to get -232.

$$7x-59y=-232$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$36y+48=4\left(8x-3y\right)+3\left(9y-5x\right)$$

Use the distributive property to multiply 4 by 8x-3y.

$$36y+48=32x-12y+3\left(9y-5x\right)$$

Use the distributive property to multiply 3 by 9y-5x.

$$36y+48=32x-12y+27y-15x$$

Combine -12y and 27y to get 15y.

$$36y+48=32x+15y-15x$$

Combine 32x and -15x to get 17x.

$$36y+48=17x+15y$$

Subtract 17x from both sides.

$$36y+48-17x=15y$$

Subtract 15y from both sides.

$$36y+48-17x-15y=0$$

Combine 36y and -15y to get 21y.

$$21y+48-17x=0$$

Subtract 48 from both sides. Anything subtracted from zero gives its negation.

$$21y-17x=-48$$

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.

$$7x-59y=-232,-17x+21y=-48$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$7x-59y=-232$$

Add 59y to both sides of the equation.

$$7x=59y-232$$

Divide both sides by 7.

$$x=\frac{1}{7}\left(59y-232\right)$$

Multiply \frac{1}{7} times 59y-232.

$$x=\frac{59}{7}y-\frac{232}{7}$$

Substitute \frac{59y-232}{7} for x in the other equation, -17x+21y=-48.

$$-17\left(\frac{59}{7}y-\frac{232}{7}\right)+21y=-48$$

Multiply -17 times \frac{59y-232}{7}.

$$-\frac{1003}{7}y+\frac{3944}{7}+21y=-48$$

Add -\frac{1003y}{7} to 21y.

$$-\frac{856}{7}y+\frac{3944}{7}=-48$$

Subtract \frac{3944}{7} from both sides of the equation.

$$-\frac{856}{7}y=-\frac{4280}{7}$$

Divide both sides of the equation by -\frac{856}{7}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=5$$

Substitute 5 for y in x=\frac{59}{7}y-\frac{232}{7}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{59}{7}\times 5-\frac{232}{7}$$

Multiply \frac{59}{7} times 5.

$$x=\frac{295-232}{7}$$

Add -\frac{232}{7} to \frac{295}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=9$$

The system is now solved.

$$x=9,y=5$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 28, the least common multiple of 4,7.

$$7\left(x-y\right)+4\left(y+2\right)=56y-224$$

Use the distributive property to multiply 7 by x-y.

$$7x-7y+4\left(y+2\right)=56y-224$$

Use the distributive property to multiply 4 by y+2.

$$7x-7y+4y+8=56y-224$$

Combine -7y and 4y to get -3y.

$$7x-3y+8=56y-224$$

Subtract 56y from both sides.

$$7x-3y+8-56y=-224$$

Combine -3y and -56y to get -59y.

$$7x-59y+8=-224$$

Subtract 8 from both sides.

$$7x-59y=-224-8$$

Subtract 8 from -224 to get -232.

$$7x-59y=-232$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$36y+48=4\left(8x-3y\right)+3\left(9y-5x\right)$$

Use the distributive property to multiply 4 by 8x-3y.

$$36y+48=32x-12y+3\left(9y-5x\right)$$

Use the distributive property to multiply 3 by 9y-5x.

$$36y+48=32x-12y+27y-15x$$

Combine -12y and 27y to get 15y.

$$36y+48=32x+15y-15x$$

Combine 32x and -15x to get 17x.

$$36y+48=17x+15y$$

Subtract 17x from both sides.

$$36y+48-17x=15y$$

Subtract 15y from both sides.

$$36y+48-17x-15y=0$$

Combine 36y and -15y to get 21y.

$$21y+48-17x=0$$

Subtract 48 from both sides. Anything subtracted from zero gives its negation.

$$21y-17x=-48$$

Put the equations in standard form and then use matrices to solve the system of equations.

$$7x-59y=-232,-17x+21y=-48$$

Write the equations in matrix form.

$$\left(\begin{matrix}7&-59\\-17&21\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-232\\-48\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}7&-59\\-17&21\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-59\\-17&21\end{matrix}\right))\left(\begin{matrix}7&-59\\-17&21\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-59\\-17&21\end{matrix}\right))\left(\begin{matrix}-232\\-48\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-59\\-17&21\end{matrix}\right))\left(\begin{matrix}-232\\-48\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-59\\-17&21\end{matrix}\right))\left(\begin{matrix}-232\\-48\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{21}{7\times 21-\left(-59\left(-17\right)\right)}&-\frac{-59}{7\times 21-\left(-59\left(-17\right)\right)}\\-\frac{-17}{7\times 21-\left(-59\left(-17\right)\right)}&\frac{7}{7\times 21-\left(-59\left(-17\right)\right)}\end{matrix}\right)\left(\begin{matrix}-232\\-48\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{21}{856}&-\frac{59}{856}\\-\frac{17}{856}&-\frac{7}{856}\end{matrix}\right)\left(\begin{matrix}-232\\-48\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{21}{856}\left(-232\right)-\frac{59}{856}\left(-48\right)\\-\frac{17}{856}\left(-232\right)-\frac{7}{856}\left(-48\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\5\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=9,y=5$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 28, the least common multiple of 4,7.

$$7\left(x-y\right)+4\left(y+2\right)=56y-224$$

Use the distributive property to multiply 7 by x-y.

$$7x-7y+4\left(y+2\right)=56y-224$$

Use the distributive property to multiply 4 by y+2.

$$7x-7y+4y+8=56y-224$$

Combine -7y and 4y to get -3y.

$$7x-3y+8=56y-224$$

Subtract 56y from both sides.

$$7x-3y+8-56y=-224$$

Combine -3y and -56y to get -59y.

$$7x-59y+8=-224$$

Subtract 8 from both sides.

$$7x-59y=-224-8$$

Subtract 8 from -224 to get -232.

$$7x-59y=-232$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$36y+48=4\left(8x-3y\right)+3\left(9y-5x\right)$$

Use the distributive property to multiply 4 by 8x-3y.

$$36y+48=32x-12y+3\left(9y-5x\right)$$

Use the distributive property to multiply 3 by 9y-5x.

$$36y+48=32x-12y+27y-15x$$

Combine -12y and 27y to get 15y.

$$36y+48=32x+15y-15x$$

Combine 32x and -15x to get 17x.

$$36y+48=17x+15y$$

Subtract 17x from both sides.

$$36y+48-17x=15y$$

Subtract 15y from both sides.

$$36y+48-17x-15y=0$$

Combine 36y and -15y to get 21y.

$$21y+48-17x=0$$

Subtract 48 from both sides. Anything subtracted from zero gives its negation.

$$21y-17x=-48$$

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.

$$7x-59y=-232,-17x+21y=-48$$

To make 7x and -17x equal, multiply all terms on each side of the first equation by -17 and all terms on each side of the second by 7.

$$-17\times 7x-17\left(-59\right)y=-17\left(-232\right),7\left(-17\right)x+7\times 21y=7\left(-48\right)$$

Simplify.

$$-119x+1003y=3944,-119x+147y=-336$$

Subtract -119x+147y=-336 from -119x+1003y=3944 by subtracting like terms on each side of the equal sign.

$$-119x+119x+1003y-147y=3944+336$$

Add -119x to 119x. Terms -119x and 119x cancel out, leaving an equation with only one variable that can be solved.

$$1003y-147y=3944+336$$

Add 1003y to -147y.

$$856y=3944+336$$

Add 3944 to 336.

$$856y=4280$$

Divide both sides by 856.

$$y=5$$

Substitute 5 for y in -17x+21y=-48. Because the resulting equation contains only one variable, you can solve for x directly.

$$-17x+21\times 5=-48$$

Multiply 21 times 5.

$$-17x+105=-48$$

Subtract 105 from both sides of the equation.

$$-17x=-153$$

Divide both sides by -17.

$$x=9$$

The system is now solved.

$$x=9,y=5$$