Solve {l}{y=x+25}{6x+6y=66}

$$\left. \begin{array} { l } { y = x + 25 } \\ { 6 x + 6 y = 66 } \end{array} \right.$$

Solve for y, x

$$x=-7$$

$$y=18$$

Steps Using Substitution

Consider the first equation. Subtract x from both sides.

$$y-x=25$$

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.

$$y-x=25,6y+6x=66$$

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

$$y-x=25$$

Add x to both sides of the equation.

$$y=x+25$$

Substitute x+25 for y in the other equation, 6y+6x=66.

$$6\left(x+25\right)+6x=66$$

Multiply 6 times x+25.

$$6x+150+6x=66$$

Add 6x to 6x.

$$12x+150=66$$

Subtract 150 from both sides of the equation.

$$12x=-84$$

Divide both sides by 12.

$$x=-7$$

Substitute -7 for x in y=x+25. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=-7+25$$

Add 25 to -7.

$$y=18$$

The system is now solved.

$$y=18,x=-7$$

Steps Using Matrices

Consider the first equation. Subtract x from both sides.

$$y-x=25$$

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

$$y-x=25,6y+6x=66$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-1\\6&6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}25\\66\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-1\\6&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\6&6\end{matrix}\right))\left(\begin{matrix}1&-1\\6&6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\6&6\end{matrix}\right))\left(\begin{matrix}25\\66\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\6&6\end{matrix}\right))\left(\begin{matrix}25\\66\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\6&6\end{matrix}\right))\left(\begin{matrix}25\\66\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{6}{6-\left(-6\right)}&-\frac{-1}{6-\left(-6\right)}\\-\frac{6}{6-\left(-6\right)}&\frac{1}{6-\left(-6\right)}\end{matrix}\right)\left(\begin{matrix}25\\66\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{12}\\-\frac{1}{2}&\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}25\\66\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 25+\frac{1}{12}\times 66\\-\frac{1}{2}\times 25+\frac{1}{12}\times 66\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}18\\-7\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=18,x=-7$$

Steps Using Elimination

Consider the first equation. Subtract x from both sides.

$$y-x=25$$

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.

$$y-x=25,6y+6x=66$$

To make y and 6y equal, multiply all terms on each side of the first equation by 6 and all terms on each side of the second by 1.

$$6y+6\left(-1\right)x=6\times 25,6y+6x=66$$

Simplify.

$$6y-6x=150,6y+6x=66$$

Subtract 6y+6x=66 from 6y-6x=150 by subtracting like terms on each side of the equal sign.

$$6y-6y-6x-6x=150-66$$

Add 6y to -6y. Terms 6y and -6y cancel out, leaving an equation with only one variable that can be solved.

$$-6x-6x=150-66$$

Add -6x to -6x.

$$-12x=150-66$$

Add 150 to -66.

$$-12x=84$$

Divide both sides by -12.

$$x=-7$$

Substitute -7 for x in 6y+6x=66. Because the resulting equation contains only one variable, you can solve for y directly.

$$6y+6\left(-7\right)=66$$

Multiply 6 times -7.

$$6y-42=66$$

Add 42 to both sides of the equation.

$$6y=108$$

Divide both sides by 6.

$$y=18$$

The system is now solved.

$$y=18,x=-7$$