Solve 3x+9=6ytext{and}-2x-2y-12=0

$$3 x + 9 = 6 y \text { and } - 2 x - 2 y - 12 = 0$$

Solve for x, y

$$x=-5$$

$$y=-1$$

Steps Using Substitution

Consider the first equation. Subtract 6y from both sides.

$$3x+9-6y=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$3x-6y=-9$$

Consider the second equation. Add 12 to both sides. Anything plus zero gives itself.

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

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.

$$3x-6y=-9,-2x-2y=12$$

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

$$3x-6y=-9$$

Add 6y to both sides of the equation.

$$3x=6y-9$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(6y-9\right)$$

Multiply \frac{1}{3} times 6y-9.

$$x=2y-3$$

Substitute 2y-3 for x in the other equation, -2x-2y=12.

$$-2\left(2y-3\right)-2y=12$$

Multiply -2 times 2y-3.

$$-4y+6-2y=12$$

Add -4y to -2y.

$$-6y+6=12$$

Subtract 6 from both sides of the equation.

$$-6y=6$$

Divide both sides by -6.

$$y=-1$$

Substitute -1 for y in x=2y-3. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=2\left(-1\right)-3$$

Multiply 2 times -1.

$$x=-2-3$$

Add -3 to -2.

$$x=-5$$

The system is now solved.

$$x=-5,y=-1$$

Steps Using Matrices

Consider the first equation. Subtract 6y from both sides.

$$3x+9-6y=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$3x-6y=-9$$

Consider the second equation. Add 12 to both sides. Anything plus zero gives itself.

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

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

$$3x-6y=-9,-2x-2y=12$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\12\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\-2&-2\end{matrix}\right))\left(\begin{matrix}-9\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3\left(-2\right)-\left(-6\left(-2\right)\right)}&-\frac{-6}{3\left(-2\right)-\left(-6\left(-2\right)\right)}\\-\frac{-2}{3\left(-2\right)-\left(-6\left(-2\right)\right)}&\frac{3}{3\left(-2\right)-\left(-6\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}-9\\12\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&-\frac{1}{3}\\-\frac{1}{9}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-9\\12\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\left(-9\right)-\frac{1}{3}\times 12\\-\frac{1}{9}\left(-9\right)-\frac{1}{6}\times 12\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\-1\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-5,y=-1$$

Steps Using Elimination

Consider the first equation. Subtract 6y from both sides.

$$3x+9-6y=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$3x-6y=-9$$

Consider the second equation. Add 12 to both sides. Anything plus zero gives itself.

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

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.

$$3x-6y=-9,-2x-2y=12$$

To make 3x and -2x equal, multiply all terms on each side of the first equation by -2 and all terms on each side of the second by 3.

$$-2\times 3x-2\left(-6\right)y=-2\left(-9\right),3\left(-2\right)x+3\left(-2\right)y=3\times 12$$

Simplify.

$$-6x+12y=18,-6x-6y=36$$

Subtract -6x-6y=36 from -6x+12y=18 by subtracting like terms on each side of the equal sign.

$$-6x+6x+12y+6y=18-36$$

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

$$12y+6y=18-36$$

Add 12y to 6y.

$$18y=18-36$$

Add 18 to -36.

$$18y=-18$$

Divide both sides by 18.

$$y=-1$$

Substitute -1 for y in -2x-2y=12. Because the resulting equation contains only one variable, you can solve for x directly.

$$-2x-2\left(-1\right)=12$$

Multiply -2 times -1.

$$-2x+2=12$$

Subtract 2 from both sides of the equation.

$$-2x=10$$

Divide both sides by -2.

$$x=-5$$

The system is now solved.

$$x=-5,y=-1$$