Solve {l}{x+8/8-y+16/9=9}{x+y/17=x-y/8-frac{17}{8}}

$$\left. \begin{array} { l } { \frac { x + 8 } { 8 } - \frac { y + 16 } { 9 } = 9 } \\ { \frac { x + y } { 17 } = \frac { x - y } { 8 } - \frac { 17 } { 8 } } \end{array} \right.$$

Solve for x, y

$$x = \frac{5096}{51} = 99\frac{47}{51} \approx 99.921568627$$

$$y = \frac{415}{17} = 24\frac{7}{17} \approx 24.411764706$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 72, the least common multiple of 8,9.

$$9\left(x+8\right)-8\left(y+16\right)=648$$

Use the distributive property to multiply 9 by x+8.

$$9x+72-8\left(y+16\right)=648$$

Use the distributive property to multiply -8 by y+16.

$$9x+72-8y-128=648$$

Subtract 128 from 72 to get -56.

$$9x-56-8y=648$$

Add 56 to both sides.

$$9x-8y=648+56$$

Add 648 and 56 to get 704.

$$9x-8y=704$$

Consider the second equation. Multiply both sides of the equation by 136, the least common multiple of 17,8.

$$8\left(x+y\right)=17\left(x-y\right)-289$$

Use the distributive property to multiply 8 by x+y.

$$8x+8y=17\left(x-y\right)-289$$

Use the distributive property to multiply 17 by x-y.

$$8x+8y=17x-17y-289$$

Subtract 17x from both sides.

$$8x+8y-17x=-17y-289$$

Combine 8x and -17x to get -9x.

$$-9x+8y=-17y-289$$

Add 17y to both sides.

$$-9x+8y+17y=-289$$

Combine 8y and 17y to get 25y.

$$-9x+25y=-289$$

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.

$$9x-8y=704,-9x+25y=-289$$

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

$$9x-8y=704$$

Add 8y to both sides of the equation.

$$9x=8y+704$$

Divide both sides by 9.

$$x=\frac{1}{9}\left(8y+704\right)$$

Multiply \frac{1}{9} times 704+8y.

$$x=\frac{8}{9}y+\frac{704}{9}$$

Substitute \frac{704+8y}{9} for x in the other equation, -9x+25y=-289.

$$-9\left(\frac{8}{9}y+\frac{704}{9}\right)+25y=-289$$

Multiply -9 times \frac{704+8y}{9}.

$$-8y-704+25y=-289$$

Add -8y to 25y.

$$17y-704=-289$$

Add 704 to both sides of the equation.

$$17y=415$$

Divide both sides by 17.

$$y=\frac{415}{17}$$

Substitute \frac{415}{17} for y in x=\frac{8}{9}y+\frac{704}{9}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{8}{9}\times \frac{415}{17}+\frac{704}{9}$$

Multiply \frac{8}{9} times \frac{415}{17} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{3320}{153}+\frac{704}{9}$$

Add \frac{704}{9} to \frac{3320}{153} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{5096}{51}$$

The system is now solved.

$$x=\frac{5096}{51},y=\frac{415}{17}$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 72, the least common multiple of 8,9.

$$9\left(x+8\right)-8\left(y+16\right)=648$$

Use the distributive property to multiply 9 by x+8.

$$9x+72-8\left(y+16\right)=648$$

Use the distributive property to multiply -8 by y+16.

$$9x+72-8y-128=648$$

Subtract 128 from 72 to get -56.

$$9x-56-8y=648$$

Add 56 to both sides.

$$9x-8y=648+56$$

Add 648 and 56 to get 704.

$$9x-8y=704$$

Consider the second equation. Multiply both sides of the equation by 136, the least common multiple of 17,8.

$$8\left(x+y\right)=17\left(x-y\right)-289$$

Use the distributive property to multiply 8 by x+y.

$$8x+8y=17\left(x-y\right)-289$$

Use the distributive property to multiply 17 by x-y.

$$8x+8y=17x-17y-289$$

Subtract 17x from both sides.

$$8x+8y-17x=-17y-289$$

Combine 8x and -17x to get -9x.

$$-9x+8y=-17y-289$$

Add 17y to both sides.

$$-9x+8y+17y=-289$$

Combine 8y and 17y to get 25y.

$$-9x+25y=-289$$

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

$$9x-8y=704,-9x+25y=-289$$

Write the equations in matrix form.

$$\left(\begin{matrix}9&-8\\-9&25\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}704\\-289\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}9&-8\\-9&25\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-8\\-9&25\end{matrix}\right))\left(\begin{matrix}9&-8\\-9&25\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-8\\-9&25\end{matrix}\right))\left(\begin{matrix}704\\-289\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-8\\-9&25\end{matrix}\right))\left(\begin{matrix}704\\-289\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-8\\-9&25\end{matrix}\right))\left(\begin{matrix}704\\-289\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{9\times 25-\left(-8\left(-9\right)\right)}&-\frac{-8}{9\times 25-\left(-8\left(-9\right)\right)}\\-\frac{-9}{9\times 25-\left(-8\left(-9\right)\right)}&\frac{9}{9\times 25-\left(-8\left(-9\right)\right)}\end{matrix}\right)\left(\begin{matrix}704\\-289\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{153}&\frac{8}{153}\\\frac{1}{17}&\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}704\\-289\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{153}\times 704+\frac{8}{153}\left(-289\right)\\\frac{1}{17}\times 704+\frac{1}{17}\left(-289\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5096}{51}\\\frac{415}{17}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{5096}{51},y=\frac{415}{17}$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 72, the least common multiple of 8,9.

$$9\left(x+8\right)-8\left(y+16\right)=648$$

Use the distributive property to multiply 9 by x+8.

$$9x+72-8\left(y+16\right)=648$$

Use the distributive property to multiply -8 by y+16.

$$9x+72-8y-128=648$$

Subtract 128 from 72 to get -56.

$$9x-56-8y=648$$

Add 56 to both sides.

$$9x-8y=648+56$$

Add 648 and 56 to get 704.

$$9x-8y=704$$

Consider the second equation. Multiply both sides of the equation by 136, the least common multiple of 17,8.

$$8\left(x+y\right)=17\left(x-y\right)-289$$

Use the distributive property to multiply 8 by x+y.

$$8x+8y=17\left(x-y\right)-289$$

Use the distributive property to multiply 17 by x-y.

$$8x+8y=17x-17y-289$$

Subtract 17x from both sides.

$$8x+8y-17x=-17y-289$$

Combine 8x and -17x to get -9x.

$$-9x+8y=-17y-289$$

Add 17y to both sides.

$$-9x+8y+17y=-289$$

Combine 8y and 17y to get 25y.

$$-9x+25y=-289$$

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.

$$9x-8y=704,-9x+25y=-289$$

To make 9x 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 9.

$$-9\times 9x-9\left(-8\right)y=-9\times 704,9\left(-9\right)x+9\times 25y=9\left(-289\right)$$

Simplify.

$$-81x+72y=-6336,-81x+225y=-2601$$

Subtract -81x+225y=-2601 from -81x+72y=-6336 by subtracting like terms on each side of the equal sign.

$$-81x+81x+72y-225y=-6336+2601$$

Add -81x to 81x. Terms -81x and 81x cancel out, leaving an equation with only one variable that can be solved.

$$72y-225y=-6336+2601$$

Add 72y to -225y.

$$-153y=-6336+2601$$

Add -6336 to 2601.

$$-153y=-3735$$

Divide both sides by -153.

$$y=\frac{415}{17}$$

Substitute \frac{415}{17} for y in -9x+25y=-289. Because the resulting equation contains only one variable, you can solve for x directly.

$$-9x+25\times \frac{415}{17}=-289$$

Multiply 25 times \frac{415}{17}.

$$-9x+\frac{10375}{17}=-289$$

Subtract \frac{10375}{17} from both sides of the equation.

$$-9x=-\frac{15288}{17}$$

Divide both sides by -9.

$$x=\frac{5096}{51}$$

The system is now solved.

$$x=\frac{5096}{51},y=\frac{415}{17}$$