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

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

Solve for x, y

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$y = -\frac{13}{10} = -1\frac{3}{10} = -1.3$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 3 by x+y.

$$3x+3y+9=2\left(x-y\right)$$

Use the distributive property to multiply 2 by x-y.

$$3x+3y+9=2x-2y$$

Subtract 2x from both sides.

$$3x+3y+9-2x=-2y$$

Combine 3x and -2x to get x.

$$x+3y+9=-2y$$

Add 2y to both sides.

$$x+3y+9+2y=0$$

Combine 3y and 2y to get 5y.

$$x+5y+9=0$$

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

$$x+5y=-9$$

Consider the second equation. Use the distributive property to multiply 2 by x+y.

$$2x+2y=3\left(x-y\right)-4$$

Use the distributive property to multiply 3 by x-y.

$$2x+2y=3x-3y-4$$

Subtract 3x from both sides.

$$2x+2y-3x=-3y-4$$

Combine 2x and -3x to get -x.

$$-x+2y=-3y-4$$

Add 3y to both sides.

$$-x+2y+3y=-4$$

Combine 2y and 3y to get 5y.

$$-x+5y=-4$$

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.

$$x+5y=-9,-x+5y=-4$$

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

$$x+5y=-9$$

Subtract 5y from both sides of the equation.

$$x=-5y-9$$

Substitute -5y-9 for x in the other equation, -x+5y=-4.

$$-\left(-5y-9\right)+5y=-4$$

Multiply -1 times -5y-9.

$$5y+9+5y=-4$$

Add 5y to 5y.

$$10y+9=-4$$

Subtract 9 from both sides of the equation.

$$10y=-13$$

Divide both sides by 10.

$$y=-\frac{13}{10}$$

Substitute -\frac{13}{10} for y in x=-5y-9. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-5\left(-\frac{13}{10}\right)-9$$

Multiply -5 times -\frac{13}{10}.

$$x=\frac{13}{2}-9$$

Add -9 to \frac{13}{2}.

$$x=-\frac{5}{2}$$

The system is now solved.

$$x=-\frac{5}{2},y=-\frac{13}{10}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 3 by x+y.

$$3x+3y+9=2\left(x-y\right)$$

Use the distributive property to multiply 2 by x-y.

$$3x+3y+9=2x-2y$$

Subtract 2x from both sides.

$$3x+3y+9-2x=-2y$$

Combine 3x and -2x to get x.

$$x+3y+9=-2y$$

Add 2y to both sides.

$$x+3y+9+2y=0$$

Combine 3y and 2y to get 5y.

$$x+5y+9=0$$

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

$$x+5y=-9$$

Consider the second equation. Use the distributive property to multiply 2 by x+y.

$$2x+2y=3\left(x-y\right)-4$$

Use the distributive property to multiply 3 by x-y.

$$2x+2y=3x-3y-4$$

Subtract 3x from both sides.

$$2x+2y-3x=-3y-4$$

Combine 2x and -3x to get -x.

$$-x+2y=-3y-4$$

Add 3y to both sides.

$$-x+2y+3y=-4$$

Combine 2y and 3y to get 5y.

$$-x+5y=-4$$

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

$$x+5y=-9,-x+5y=-4$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&5\\-1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&5\\-1&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&5\\-1&5\end{matrix}\right))\left(\begin{matrix}1&5\\-1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\-1&5\end{matrix}\right))\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\-1&5\end{matrix}\right))\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&5\\-1&5\end{matrix}\right))\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5-5\left(-1\right)}&-\frac{5}{5-5\left(-1\right)}\\-\frac{-1}{5-5\left(-1\right)}&\frac{1}{5-5\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\\frac{1}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}-9\\-4\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\left(-9\right)-\frac{1}{2}\left(-4\right)\\\frac{1}{10}\left(-9\right)+\frac{1}{10}\left(-4\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2}\\-\frac{13}{10}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{5}{2},y=-\frac{13}{10}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 3 by x+y.

$$3x+3y+9=2\left(x-y\right)$$

Use the distributive property to multiply 2 by x-y.

$$3x+3y+9=2x-2y$$

Subtract 2x from both sides.

$$3x+3y+9-2x=-2y$$

Combine 3x and -2x to get x.

$$x+3y+9=-2y$$

Add 2y to both sides.

$$x+3y+9+2y=0$$

Combine 3y and 2y to get 5y.

$$x+5y+9=0$$

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

$$x+5y=-9$$

Consider the second equation. Use the distributive property to multiply 2 by x+y.

$$2x+2y=3\left(x-y\right)-4$$

Use the distributive property to multiply 3 by x-y.

$$2x+2y=3x-3y-4$$

Subtract 3x from both sides.

$$2x+2y-3x=-3y-4$$

Combine 2x and -3x to get -x.

$$-x+2y=-3y-4$$

Add 3y to both sides.

$$-x+2y+3y=-4$$

Combine 2y and 3y to get 5y.

$$-x+5y=-4$$

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.

$$x+5y=-9,-x+5y=-4$$

Subtract -x+5y=-4 from x+5y=-9 by subtracting like terms on each side of the equal sign.

$$x+x+5y-5y=-9+4$$

Add 5y to -5y. Terms 5y and -5y cancel out, leaving an equation with only one variable that can be solved.

$$x+x=-9+4$$

Add x to x.

$$2x=-9+4$$

Add -9 to 4.

$$2x=-5$$

Divide both sides by 2.

$$x=-\frac{5}{2}$$

Substitute -\frac{5}{2} for x in -x+5y=-4. Because the resulting equation contains only one variable, you can solve for y directly.

$$-\left(-\frac{5}{2}\right)+5y=-4$$

Multiply -1 times -\frac{5}{2}.

$$\frac{5}{2}+5y=-4$$

Subtract \frac{5}{2} from both sides of the equation.

$$5y=-\frac{13}{2}$$

Divide both sides by 5.

$$y=-\frac{13}{10}$$

The system is now solved.

$$x=-\frac{5}{2},y=-\frac{13}{10}$$