Solve {l}{2(x+1)+3y=7}{3x+2y-5=-10}

$$\left\{ \begin{array} { l } { 2 ( x + 1 ) + 3 y = 7 } \\ { 3 x + 2 y - 5 = - 10 } \end{array} \right.$$

Solve for x, y

$$x=-5$$

$$y=5$$

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.

$$2\left(x+1\right)+3y=7,3x+2y-5=-10$$

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

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

Multiply 2 times x+1.

$$2x+2+3y=7$$

Subtract 2 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, 3x+2y-5=-10.

$$3\left(-\frac{3}{2}y+\frac{5}{2}\right)+2y-5=-10$$

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

$$-\frac{9}{2}y+\frac{15}{2}+2y-5=-10$$

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

$$-\frac{5}{2}y+\frac{15}{2}-5=-10$$

Add \frac{15}{2} to -5.

$$-\frac{5}{2}y+\frac{5}{2}=-10$$

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

$$-\frac{5}{2}y=-\frac{25}{2}$$

Divide both sides of the equation by -\frac{5}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=5$$

Substitute 5 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.

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

Multiply -\frac{3}{2} times 5.

$$x=\frac{-15+5}{2}$$

Add \frac{5}{2} to -\frac{15}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-5$$

The system is now solved.

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

Steps Using Matrices

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

$$2\left(x+1\right)+3y=7,3x+2y-5=-10$$

Simplify the first equation to put it in standard form.

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

Multiply 2 times x+1.

$$2x+2+3y=7$$

Subtract 2 from both sides of the equation.

$$2x+3y=5$$

Write the equations in matrix form.

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

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

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

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

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

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

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

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.

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

Do the arithmetic.

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

Multiply the matrices.

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

Do the arithmetic.

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

Extract the matrix elements x and y.

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