Solve {l}{18x+22y=2,264}{7x+15y=1,338}

$$\left. \begin{array} { l } { 18 x + 22 y = 2,264 } \\ { 7 x + 15 y = 1,338 } \end{array} \right.$$

Solve for x, y

$$x=39$$

$$y=71$$

Steps Using Substitution

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.

$$18x+22y=2264,7x+15y=1338$$

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

$$18x+22y=2264$$

Subtract 22y from both sides of the equation.

$$18x=-22y+2264$$

Divide both sides by 18.

$$x=\frac{1}{18}\left(-22y+2264\right)$$

Multiply \frac{1}{18} times -22y+2264.

$$x=-\frac{11}{9}y+\frac{1132}{9}$$

Substitute \frac{-11y+1132}{9} for x in the other equation, 7x+15y=1338.

$$7\left(-\frac{11}{9}y+\frac{1132}{9}\right)+15y=1338$$

Multiply 7 times \frac{-11y+1132}{9}.

$$-\frac{77}{9}y+\frac{7924}{9}+15y=1338$$

Add -\frac{77y}{9} to 15y.

$$\frac{58}{9}y+\frac{7924}{9}=1338$$

Subtract \frac{7924}{9} from both sides of the equation.

$$\frac{58}{9}y=\frac{4118}{9}$$

Divide both sides of the equation by \frac{58}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=71$$

Substitute 71 for y in x=-\frac{11}{9}y+\frac{1132}{9}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{11}{9}\times 71+\frac{1132}{9}$$

Multiply -\frac{11}{9} times 71.

$$x=\frac{-781+1132}{9}$$

Add \frac{1132}{9} to -\frac{781}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=39$$

The system is now solved.

$$x=39,y=71$$

Steps Using Matrices

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

$$18x+22y=2264,7x+15y=1338$$

Write the equations in matrix form.

$$\left(\begin{matrix}18&22\\7&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2264\\1338\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}18&22\\7&15\end{matrix}\right).

$$inverse(\left(\begin{matrix}18&22\\7&15\end{matrix}\right))\left(\begin{matrix}18&22\\7&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&22\\7&15\end{matrix}\right))\left(\begin{matrix}2264\\1338\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&22\\7&15\end{matrix}\right))\left(\begin{matrix}2264\\1338\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}18&22\\7&15\end{matrix}\right))\left(\begin{matrix}2264\\1338\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{18\times 15-22\times 7}&-\frac{22}{18\times 15-22\times 7}\\-\frac{7}{18\times 15-22\times 7}&\frac{18}{18\times 15-22\times 7}\end{matrix}\right)\left(\begin{matrix}2264\\1338\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{116}&-\frac{11}{58}\\-\frac{7}{116}&\frac{9}{58}\end{matrix}\right)\left(\begin{matrix}2264\\1338\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{116}\times 2264-\frac{11}{58}\times 1338\\-\frac{7}{116}\times 2264+\frac{9}{58}\times 1338\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}39\\71\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=39,y=71$$

Steps Using Elimination

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.

$$18x+22y=2264,7x+15y=1338$$

To make 18x and 7x equal, multiply all terms on each side of the first equation by 7 and all terms on each side of the second by 18.

$$7\times 18x+7\times 22y=7\times 2264,18\times 7x+18\times 15y=18\times 1338$$

Simplify.

$$126x+154y=15848,126x+270y=24084$$

Subtract 126x+270y=24084 from 126x+154y=15848 by subtracting like terms on each side of the equal sign.

$$126x-126x+154y-270y=15848-24084$$

Add 126x to -126x. Terms 126x and -126x cancel out, leaving an equation with only one variable that can be solved.

$$154y-270y=15848-24084$$

Add 154y to -270y.

$$-116y=15848-24084$$

Add 15848 to -24084.

$$-116y=-8236$$

Divide both sides by -116.

$$y=71$$

Substitute 71 for y in 7x+15y=1338. Because the resulting equation contains only one variable, you can solve for x directly.

$$7x+15\times 71=1338$$

Multiply 15 times 71.

$$7x+1065=1338$$

Subtract 1065 from both sides of the equation.

$$7x=273$$

Divide both sides by 7.

$$x=39$$

The system is now solved.

$$x=39,y=71$$