Solve {l}{2x+12y=-70}{10x=-6y+28}

$$\left\{ \begin{array} { l } { 2 x + 12 y = - 70 } \\ { 10 x = - 6 y + 28 } \end{array} \right.$$

Solve for x, y

$$x=7$$

$$y=-7$$

Steps Using Substitution

Consider the second equation. Add 6y to both sides.

$$10x+6y=28$$

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.

$$2x+12y=-70,10x+6y=28$$

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

$$2x+12y=-70$$

Subtract 12y from both sides of the equation.

$$2x=-12y-70$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-12y-70\right)$$

Multiply \frac{1}{2} times -12y-70.

$$x=-6y-35$$

Substitute -6y-35 for x in the other equation, 10x+6y=28.

$$10\left(-6y-35\right)+6y=28$$

Multiply 10 times -6y-35.

$$-60y-350+6y=28$$

Add -60y to 6y.

$$-54y-350=28$$

Add 350 to both sides of the equation.

$$-54y=378$$

Divide both sides by -54.

$$y=-7$$

Substitute -7 for y in x=-6y-35. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-6\left(-7\right)-35$$

Multiply -6 times -7.

$$x=42-35$$

Add -35 to 42.

$$x=7$$

The system is now solved.

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

Steps Using Matrices

Consider the second equation. Add 6y to both sides.

$$10x+6y=28$$

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

$$2x+12y=-70,10x+6y=28$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&12\\10&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-70\\28\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&12\\10&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&12\\10&6\end{matrix}\right))\left(\begin{matrix}2&12\\10&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&12\\10&6\end{matrix}\right))\left(\begin{matrix}-70\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&12\\10&6\end{matrix}\right))\left(\begin{matrix}-70\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&12\\10&6\end{matrix}\right))\left(\begin{matrix}-70\\28\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{2\times 6-12\times 10}&-\frac{12}{2\times 6-12\times 10}\\-\frac{10}{2\times 6-12\times 10}&\frac{2}{2\times 6-12\times 10}\end{matrix}\right)\left(\begin{matrix}-70\\28\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{18}&\frac{1}{9}\\\frac{5}{54}&-\frac{1}{54}\end{matrix}\right)\left(\begin{matrix}-70\\28\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{18}\left(-70\right)+\frac{1}{9}\times 28\\\frac{5}{54}\left(-70\right)-\frac{1}{54}\times 28\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

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

Steps Using Elimination

Consider the second equation. Add 6y to both sides.

$$10x+6y=28$$

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.

$$2x+12y=-70,10x+6y=28$$

To make 2x 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 2.

$$10\times 2x+10\times 12y=10\left(-70\right),2\times 10x+2\times 6y=2\times 28$$

Simplify.

$$20x+120y=-700,20x+12y=56$$

Subtract 20x+12y=56 from 20x+120y=-700 by subtracting like terms on each side of the equal sign.

$$20x-20x+120y-12y=-700-56$$

Add 20x to -20x. Terms 20x and -20x cancel out, leaving an equation with only one variable that can be solved.

$$120y-12y=-700-56$$

Add 120y to -12y.

$$108y=-700-56$$

Add -700 to -56.

$$108y=-756$$

Divide both sides by 108.

$$y=-7$$

Substitute -7 for y in 10x+6y=28. Because the resulting equation contains only one variable, you can solve for x directly.

$$10x+6\left(-7\right)=28$$

Multiply 6 times -7.

$$10x-42=28$$

Add 42 to both sides of the equation.

$$10x=70$$

Divide both sides by 10.

$$x=7$$

The system is now solved.

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