Solve x:y=15:13text{e}x-y=8

$$x : y = 15 : 13 \text { e } x - y = 8$$

Solve for x, y

$$x=60$$

$$y=52$$

Steps Using Substitution

Consider the first equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 13y, the least common multiple of y,13.

$$13x=15y$$

Divide both sides by 13.

$$x=\frac{1}{13}\times 15y$$

Multiply \frac{1}{13} times 15y.

$$x=\frac{15}{13}y$$

Substitute \frac{15y}{13} for x in the other equation, x-y=8.

$$\frac{15}{13}y-y=8$$

Add \frac{15y}{13} to -y.

$$\frac{2}{13}y=8$$

Divide both sides of the equation by \frac{2}{13}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=52$$

Substitute 52 for y in x=\frac{15}{13}y. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{15}{13}\times 52$$

Multiply \frac{15}{13} times 52.

$$x=60$$

The system is now solved.

$$x=60,y=52$$

Steps Using Matrices

Consider the first equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 13y, the least common multiple of y,13.

$$13x=15y$$

Subtract 15y from both sides.

$$13x-15y=0$$

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

$$13x-15y=0,x-y=8$$

Write the equations in matrix form.

$$\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\8\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}13&-15\\1&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right))\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right))\left(\begin{matrix}0\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right))\left(\begin{matrix}0\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&-15\\1&-1\end{matrix}\right))\left(\begin{matrix}0\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{13\left(-1\right)-\left(-15\right)}&-\frac{-15}{13\left(-1\right)-\left(-15\right)}\\-\frac{1}{13\left(-1\right)-\left(-15\right)}&\frac{13}{13\left(-1\right)-\left(-15\right)}\end{matrix}\right)\left(\begin{matrix}0\\8\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{15}{2}\\-\frac{1}{2}&\frac{13}{2}\end{matrix}\right)\left(\begin{matrix}0\\8\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{2}\times 8\\\frac{13}{2}\times 8\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\52\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=60,y=52$$

Steps Using Elimination

Consider the first equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 13y, the least common multiple of y,13.

$$13x=15y$$

Subtract 15y from both sides.

$$13x-15y=0$$

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.

$$13x-15y=0,x-y=8$$

To make 13x and x equal, multiply all terms on each side of the first equation by 1 and all terms on each side of the second by 13.

$$13x-15y=0,13x+13\left(-1\right)y=13\times 8$$

Simplify.

$$13x-15y=0,13x-13y=104$$

Subtract 13x-13y=104 from 13x-15y=0 by subtracting like terms on each side of the equal sign.

$$13x-13x-15y+13y=-104$$

Add 13x to -13x. Terms 13x and -13x cancel out, leaving an equation with only one variable that can be solved.

$$-15y+13y=-104$$

Add -15y to 13y.

$$-2y=-104$$

Divide both sides by -2.

$$y=52$$

Substitute 52 for y in x-y=8. Because the resulting equation contains only one variable, you can solve for x directly.

$$x-52=8$$

Add 52 to both sides of the equation.

$$x=60$$

The system is now solved.

$$x=60,y=52$$