Solve {l}{3x-2y=22}{3y+31=4x}

$$\left. \begin{array} { l } { 3 x - 2 y = 22 } \\ { 3 y + 31 = 4 x } \end{array} \right.$$

Solve for x, y

$$x=4$$

$$y=-5$$

Steps Using Substitution

Consider the second equation. Subtract 4x from both sides.

$$3y+31-4x=0$$

Subtract 31 from both sides. Anything subtracted from zero gives its negation.

$$3y-4x=-31$$

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.

$$3x-2y=22,-4x+3y=-31$$

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

$$3x-2y=22$$

Add 2y to both sides of the equation.

$$3x=2y+22$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(2y+22\right)$$

Multiply \frac{1}{3} times 22+2y.

$$x=\frac{2}{3}y+\frac{22}{3}$$

Substitute \frac{22+2y}{3} for x in the other equation, -4x+3y=-31.

$$-4\left(\frac{2}{3}y+\frac{22}{3}\right)+3y=-31$$

Multiply -4 times \frac{22+2y}{3}.

$$-\frac{8}{3}y-\frac{88}{3}+3y=-31$$

Add -\frac{8y}{3} to 3y.

$$\frac{1}{3}y-\frac{88}{3}=-31$$

Add \frac{88}{3} to both sides of the equation.

$$\frac{1}{3}y=-\frac{5}{3}$$

Multiply both sides by 3.

$$y=-5$$

Substitute -5 for y in x=\frac{2}{3}y+\frac{22}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{2}{3}\left(-5\right)+\frac{22}{3}$$

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

$$x=\frac{-10+22}{3}$$

Add \frac{22}{3} to -\frac{10}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=4$$

The system is now solved.

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

Steps Using Matrices

Consider the second equation. Subtract 4x from both sides.

$$3y+31-4x=0$$

Subtract 31 from both sides. Anything subtracted from zero gives its negation.

$$3y-4x=-31$$

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

$$3x-2y=22,-4x+3y=-31$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-2\\-4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}22\\-31\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-2\\-4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}3&-2\\-4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}22\\-31\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}22\\-31\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-4&3\end{matrix}\right))\left(\begin{matrix}22\\-31\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-2\left(-4\right)\right)}&-\frac{-2}{3\times 3-\left(-2\left(-4\right)\right)}\\-\frac{-4}{3\times 3-\left(-2\left(-4\right)\right)}&\frac{3}{3\times 3-\left(-2\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}22\\-31\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3&2\\4&3\end{matrix}\right)\left(\begin{matrix}22\\-31\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\times 22+2\left(-31\right)\\4\times 22+3\left(-31\right)\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

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

Steps Using Elimination

Consider the second equation. Subtract 4x from both sides.

$$3y+31-4x=0$$

Subtract 31 from both sides. Anything subtracted from zero gives its negation.

$$3y-4x=-31$$

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.

$$3x-2y=22,-4x+3y=-31$$

To make 3x and -4x equal, multiply all terms on each side of the first equation by -4 and all terms on each side of the second by 3.

$$-4\times 3x-4\left(-2\right)y=-4\times 22,3\left(-4\right)x+3\times 3y=3\left(-31\right)$$

Simplify.

$$-12x+8y=-88,-12x+9y=-93$$

Subtract -12x+9y=-93 from -12x+8y=-88 by subtracting like terms on each side of the equal sign.

$$-12x+12x+8y-9y=-88+93$$

Add -12x to 12x. Terms -12x and 12x cancel out, leaving an equation with only one variable that can be solved.

$$8y-9y=-88+93$$

Add 8y to -9y.

$$-y=-88+93$$

Add -88 to 93.

$$-y=5$$

Divide both sides by -1.

$$y=-5$$

Substitute -5 for y in -4x+3y=-31. Because the resulting equation contains only one variable, you can solve for x directly.

$$-4x+3\left(-5\right)=-31$$

Multiply 3 times -5.

$$-4x-15=-31$$

Add 15 to both sides of the equation.

$$-4x=-16$$

Divide both sides by -4.

$$x=4$$

The system is now solved.

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