Solve {l}{20+x+y=115}{11x=8y}

$$\left\{ \begin{array} { l } { 20 + x + y = 115 } \\ { 11 x = 8 y } \end{array} \right.$$

Solve for x, y

$$x=40$$

$$y=55$$

Steps Using Substitution

Consider the first equation. Subtract 20 from both sides.

$$x+y=115-20$$

Subtract 20 from 115 to get 95.

$$x+y=95$$

Consider the second equation. Subtract 8y from both sides.

$$11x-8y=0$$

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.

$$x+y=95,11x-8y=0$$

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

$$x+y=95$$

Subtract y from both sides of the equation.

$$x=-y+95$$

Substitute -y+95 for x in the other equation, 11x-8y=0.

$$11\left(-y+95\right)-8y=0$$

Multiply 11 times -y+95.

$$-11y+1045-8y=0$$

Add -11y to -8y.

$$-19y+1045=0$$

Subtract 1045 from both sides of the equation.

$$-19y=-1045$$

Divide both sides by -19.

$$y=55$$

Substitute 55 for y in x=-y+95. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-55+95$$

Add 95 to -55.

$$x=40$$

The system is now solved.

$$x=40,y=55$$

Steps Using Matrices

Consider the first equation. Subtract 20 from both sides.

$$x+y=115-20$$

Subtract 20 from 115 to get 95.

$$x+y=95$$

Consider the second equation. Subtract 8y from both sides.

$$11x-8y=0$$

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

$$x+y=95,11x-8y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\11&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}95\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\11&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\11&-8\end{matrix}\right))\left(\begin{matrix}1&1\\11&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&-8\end{matrix}\right))\left(\begin{matrix}95\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&-8\end{matrix}\right))\left(\begin{matrix}95\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&-8\end{matrix}\right))\left(\begin{matrix}95\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-8-11}&-\frac{1}{-8-11}\\-\frac{11}{-8-11}&\frac{1}{-8-11}\end{matrix}\right)\left(\begin{matrix}95\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{19}&\frac{1}{19}\\\frac{11}{19}&-\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}95\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{19}\times 95\\\frac{11}{19}\times 95\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\55\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=40,y=55$$

Steps Using Elimination

Consider the first equation. Subtract 20 from both sides.

$$x+y=115-20$$

Subtract 20 from 115 to get 95.

$$x+y=95$$

Consider the second equation. Subtract 8y from both sides.

$$11x-8y=0$$

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.

$$x+y=95,11x-8y=0$$

To make x and 11x equal, multiply all terms on each side of the first equation by 11 and all terms on each side of the second by 1.

$$11x+11y=11\times 95,11x-8y=0$$

Simplify.

$$11x+11y=1045,11x-8y=0$$

Subtract 11x-8y=0 from 11x+11y=1045 by subtracting like terms on each side of the equal sign.

$$11x-11x+11y+8y=1045$$

Add 11x to -11x. Terms 11x and -11x cancel out, leaving an equation with only one variable that can be solved.

$$11y+8y=1045$$

Add 11y to 8y.

$$19y=1045$$

Divide both sides by 19.

$$y=55$$

Substitute 55 for y in 11x-8y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$11x-8\times 55=0$$

Multiply -8 times 55.

$$11x-440=0$$

Add 440 to both sides of the equation.

$$11x=440$$

Divide both sides by 11.

$$x=40$$

The system is now solved.

$$x=40,y=55$$