Solve 18x+11y=522text{and}10x=112-16y

$$18 x + 11 y = 522 \text { and } 10 x = 112 - 16 y$$

Solve for x, y

$$x=40$$

$$y=-18$$

Steps Using Substitution

Consider the second equation. Add 16y to both sides.

$$10x+16y=112$$

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.

$$18x+11y=522,10x+16y=112$$

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

$$18x+11y=522$$

Subtract 11y from both sides of the equation.

$$18x=-11y+522$$

Divide both sides by 18.

$$x=\frac{1}{18}\left(-11y+522\right)$$

Multiply \frac{1}{18} times -11y+522.

$$x=-\frac{11}{18}y+29$$

Substitute -\frac{11y}{18}+29 for x in the other equation, 10x+16y=112.

$$10\left(-\frac{11}{18}y+29\right)+16y=112$$

Multiply 10 times -\frac{11y}{18}+29.

$$-\frac{55}{9}y+290+16y=112$$

Add -\frac{55y}{9} to 16y.

$$\frac{89}{9}y+290=112$$

Subtract 290 from both sides of the equation.

$$\frac{89}{9}y=-178$$

Divide both sides of the equation by \frac{89}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-18$$

Substitute -18 for y in x=-\frac{11}{18}y+29. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{11}{18}\left(-18\right)+29$$

Multiply -\frac{11}{18} times -18.

$$x=11+29$$

Add 29 to 11.

$$x=40$$

The system is now solved.

$$x=40,y=-18$$

Steps Using Matrices

Consider the second equation. Add 16y to both sides.

$$10x+16y=112$$

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

$$18x+11y=522,10x+16y=112$$

Write the equations in matrix form.

$$\left(\begin{matrix}18&11\\10&16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}522\\112\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}18&11\\10&16\end{matrix}\right).

$$inverse(\left(\begin{matrix}18&11\\10&16\end{matrix}\right))\left(\begin{matrix}18&11\\10&16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&11\\10&16\end{matrix}\right))\left(\begin{matrix}522\\112\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&11\\10&16\end{matrix}\right))\left(\begin{matrix}522\\112\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&11\\10&16\end{matrix}\right))\left(\begin{matrix}522\\112\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{18\times 16-11\times 10}&-\frac{11}{18\times 16-11\times 10}\\-\frac{10}{18\times 16-11\times 10}&\frac{18}{18\times 16-11\times 10}\end{matrix}\right)\left(\begin{matrix}522\\112\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{89}&-\frac{11}{178}\\-\frac{5}{89}&\frac{9}{89}\end{matrix}\right)\left(\begin{matrix}522\\112\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{89}\times 522-\frac{11}{178}\times 112\\-\frac{5}{89}\times 522+\frac{9}{89}\times 112\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\-18\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=40,y=-18$$

Steps Using Elimination

Consider the second equation. Add 16y to both sides.

$$10x+16y=112$$

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.

$$18x+11y=522,10x+16y=112$$

To make 18x and 10x equal, multiply all terms on each side of the first equation by 10 and all terms on each side of the second by 18.

$$10\times 18x+10\times 11y=10\times 522,18\times 10x+18\times 16y=18\times 112$$

Simplify.

$$180x+110y=5220,180x+288y=2016$$

Subtract 180x+288y=2016 from 180x+110y=5220 by subtracting like terms on each side of the equal sign.

$$180x-180x+110y-288y=5220-2016$$

Add 180x to -180x. Terms 180x and -180x cancel out, leaving an equation with only one variable that can be solved.

$$110y-288y=5220-2016$$

Add 110y to -288y.

$$-178y=5220-2016$$

Add 5220 to -2016.

$$-178y=3204$$

Divide both sides by -178.

$$y=-18$$

Substitute -18 for y in 10x+16y=112. Because the resulting equation contains only one variable, you can solve for x directly.

$$10x+16\left(-18\right)=112$$

Multiply 16 times -18.

$$10x-288=112$$

Add 288 to both sides of the equation.

$$10x=400$$

Divide both sides by 10.

$$x=40$$

The system is now solved.

$$x=40,y=-18$$