Solve {l}{2(x-2)+(3y+7)=8}{2(x-1)-3(7-3y)=24}

$$\left. \begin{array} { l } { 2 ( x - 2 ) + ( 3 y + 7 ) = 8 } \\ { 2 ( x - 1 ) - 3 ( 7 - 3 y ) = 24 } \end{array} \right.$$

Solve for x, y

$$x=-8$$

$$y=7$$

Steps Using Substitution

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.

$$2\left(x-2\right)+3y+7=8,2\left(x-1\right)-3\left(-3y+7\right)=24$$

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

$$2\left(x-2\right)+3y+7=8$$

Multiply 2 times x-2.

$$2x-4+3y+7=8$$

Add -4 to 7.

$$2x+3y+3=8$$

Subtract 3 from both sides of the equation.

$$2x+3y=5$$

Subtract 3y from both sides of the equation.

$$2x=-3y+5$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-3y+5\right)$$

Multiply \frac{1}{2} times -3y+5.

$$x=-\frac{3}{2}y+\frac{5}{2}$$

Substitute \frac{-3y+5}{2} for x in the other equation, 2\left(x-1\right)-3\left(-3y+7\right)=24.

$$2\left(-\frac{3}{2}y+\frac{5}{2}-1\right)-3\left(-3y+7\right)=24$$

Add \frac{5}{2} to -1.

$$2\left(-\frac{3}{2}y+\frac{3}{2}\right)-3\left(-3y+7\right)=24$$

Multiply 2 times \frac{-3y+3}{2}.

$$-3y+3-3\left(-3y+7\right)=24$$

Multiply -3 times -3y+7.

$$-3y+3+9y-21=24$$

Add -3y to 9y.

$$6y+3-21=24$$

Add 3 to -21.

$$6y-18=24$$

Add 18 to both sides of the equation.

$$6y=42$$

Divide both sides by 6.

$$y=7$$

Substitute 7 for y in x=-\frac{3}{2}y+\frac{5}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{2}\times 7+\frac{5}{2}$$

Multiply -\frac{3}{2} times 7.

$$x=\frac{-21+5}{2}$$

Add \frac{5}{2} to -\frac{21}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-8$$

The system is now solved.

$$x=-8,y=7$$

Steps Using Matrices

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

$$2\left(x-2\right)+3y+7=8,2\left(x-1\right)-3\left(-3y+7\right)=24$$

Simplify the first equation to put it in standard form.

$$2\left(x-2\right)+3y+7=8$$

Multiply 2 times x-2.

$$2x-4+3y+7=8$$

Add -4 to 7.

$$2x+3y+3=8$$

Subtract 3 from both sides of the equation.

$$2x+3y=5$$

Simplify the second equation to put it in standard form.

$$2\left(x-1\right)-3\left(-3y+7\right)=24$$

Multiply 2 times x-1.

$$2x-2-3\left(-3y+7\right)=24$$

Multiply -3 times -3y+7.

$$2x-2+9y-21=24$$

Add -2 to -21.

$$2x+9y-23=24$$

Add 23 to both sides of the equation.

$$2x+9y=47$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&3\\2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\47\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&3\\2&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\2&9\end{matrix}\right))\left(\begin{matrix}2&3\\2&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\2&9\end{matrix}\right))\left(\begin{matrix}5\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\2&9\end{matrix}\right))\left(\begin{matrix}5\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\2&9\end{matrix}\right))\left(\begin{matrix}5\\47\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\times 9-3\times 2}&-\frac{3}{2\times 9-3\times 2}\\-\frac{2}{2\times 9-3\times 2}&\frac{2}{2\times 9-3\times 2}\end{matrix}\right)\left(\begin{matrix}5\\47\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&-\frac{1}{4}\\-\frac{1}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}5\\47\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 5-\frac{1}{4}\times 47\\-\frac{1}{6}\times 5+\frac{1}{6}\times 47\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-8\\7\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-8,y=7$$