Solve {l}{(-3-x)(3+x)-y=-x^2}{-2/3(3/2x-9/4y)=x+2y/2}

$$\left\{ \begin{array} { l } { ( - 3 - x ) ( 3 + x ) - y = - x ^ { 2 } } \\ { - \frac { 2 } { 3 } ( \frac { 3 } { 2 } x - \frac { 9 } { 4 } y ) = \frac { x + 2 y } { 2 } } \end{array} \right.$$

Solve for x, y

$$x=-1$$

$$y=-3$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply -3-x by 3+x and combine like terms.

$$-9-6x-x^{2}-y=-x^{2}$$

Add x^{2} to both sides.

$$-9-6x-x^{2}-y+x^{2}=0$$

Combine -x^{2} and x^{2} to get 0.

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

Add 9 to both sides. Anything plus zero gives itself.

$$-6x-y=9$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 3,2,4.

$$-8\left(\frac{3}{2}x-\frac{9}{4}y\right)=6\left(x+2y\right)$$

Use the distributive property to multiply -8 by \frac{3}{2}x-\frac{9}{4}y.

$$-12x+18y=6\left(x+2y\right)$$

Use the distributive property to multiply 6 by x+2y.

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

Subtract 6x from both sides.

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

Combine -12x and -6x to get -18x.

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

Subtract 12y from both sides.

$$-18x+18y-12y=0$$

Combine 18y and -12y to get 6y.

$$-18x+6y=0$$

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.

$$-6x-y=9,-18x+6y=0$$

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

$$-6x-y=9$$

Add y to both sides of the equation.

$$-6x=y+9$$

Divide both sides by -6.

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

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

$$x=-\frac{1}{6}y-\frac{3}{2}$$

Substitute -\frac{y}{6}-\frac{3}{2} for x in the other equation, -18x+6y=0.

$$-18\left(-\frac{1}{6}y-\frac{3}{2}\right)+6y=0$$

Multiply -18 times -\frac{y}{6}-\frac{3}{2}.

$$3y+27+6y=0$$

Add 3y to 6y.

$$9y+27=0$$

Subtract 27 from both sides of the equation.

$$9y=-27$$

Divide both sides by 9.

$$y=-3$$

Substitute -3 for y in x=-\frac{1}{6}y-\frac{3}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{6}\left(-3\right)-\frac{3}{2}$$

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

$$x=\frac{1-3}{2}$$

Add -\frac{3}{2} to \frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-1$$

The system is now solved.

$$x=-1,y=-3$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply -3-x by 3+x and combine like terms.

$$-9-6x-x^{2}-y=-x^{2}$$

Add x^{2} to both sides.

$$-9-6x-x^{2}-y+x^{2}=0$$

Combine -x^{2} and x^{2} to get 0.

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

Add 9 to both sides. Anything plus zero gives itself.

$$-6x-y=9$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 3,2,4.

$$-8\left(\frac{3}{2}x-\frac{9}{4}y\right)=6\left(x+2y\right)$$

Use the distributive property to multiply -8 by \frac{3}{2}x-\frac{9}{4}y.

$$-12x+18y=6\left(x+2y\right)$$

Use the distributive property to multiply 6 by x+2y.

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

Subtract 6x from both sides.

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

Combine -12x and -6x to get -18x.

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

Subtract 12y from both sides.

$$-18x+18y-12y=0$$

Combine 18y and -12y to get 6y.

$$-18x+6y=0$$

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

$$-6x-y=9,-18x+6y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}-6&-1\\-18&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-6&-1\\-18&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&-1\\-18&6\end{matrix}\right))\left(\begin{matrix}-6&-1\\-18&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-1\\-18&6\end{matrix}\right))\left(\begin{matrix}9\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-1\\-18&6\end{matrix}\right))\left(\begin{matrix}9\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-1\\-18&6\end{matrix}\right))\left(\begin{matrix}9\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{-6\times 6-\left(-\left(-18\right)\right)}&-\frac{-1}{-6\times 6-\left(-\left(-18\right)\right)}\\-\frac{-18}{-6\times 6-\left(-\left(-18\right)\right)}&-\frac{6}{-6\times 6-\left(-\left(-18\right)\right)}\end{matrix}\right)\left(\begin{matrix}9\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{9}&-\frac{1}{54}\\-\frac{1}{3}&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}9\\0\end{matrix}\right)$$

Multiply the matrices.

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

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\-3\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-1,y=-3$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply -3-x by 3+x and combine like terms.

$$-9-6x-x^{2}-y=-x^{2}$$

Add x^{2} to both sides.

$$-9-6x-x^{2}-y+x^{2}=0$$

Combine -x^{2} and x^{2} to get 0.

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

Add 9 to both sides. Anything plus zero gives itself.

$$-6x-y=9$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 3,2,4.

$$-8\left(\frac{3}{2}x-\frac{9}{4}y\right)=6\left(x+2y\right)$$

Use the distributive property to multiply -8 by \frac{3}{2}x-\frac{9}{4}y.

$$-12x+18y=6\left(x+2y\right)$$

Use the distributive property to multiply 6 by x+2y.

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

Subtract 6x from both sides.

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

Combine -12x and -6x to get -18x.

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

Subtract 12y from both sides.

$$-18x+18y-12y=0$$

Combine 18y and -12y to get 6y.

$$-18x+6y=0$$

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.

$$-6x-y=9,-18x+6y=0$$

To make -6x and -18x equal, multiply all terms on each side of the first equation by -18 and all terms on each side of the second by -6.

$$-18\left(-6\right)x-18\left(-1\right)y=-18\times 9,-6\left(-18\right)x-6\times 6y=0$$

Simplify.

$$108x+18y=-162,108x-36y=0$$

Subtract 108x-36y=0 from 108x+18y=-162 by subtracting like terms on each side of the equal sign.

$$108x-108x+18y+36y=-162$$

Add 108x to -108x. Terms 108x and -108x cancel out, leaving an equation with only one variable that can be solved.

$$18y+36y=-162$$

Add 18y to 36y.

$$54y=-162$$

Divide both sides by 54.

$$y=-3$$

Substitute -3 for y in -18x+6y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$-18x+6\left(-3\right)=0$$

Multiply 6 times -3.

$$-18x-18=0$$

Add 18 to both sides of the equation.

$$-18x=18$$

Divide both sides by -18.

$$x=-1$$

The system is now solved.

$$x=-1,y=-3$$