Solve {l}{3x-2(y+4)=0}{4(x+1)+5y=10}

$$\left\{ \begin{array} { l } { 3 x - 2 ( y + 4 ) = 0 } \\ { 4 ( x + 1 ) + 5 y = 10 } \end{array} \right.$$

Solve for x, y

$$x = \frac{52}{23} = 2\frac{6}{23} \approx 2.260869565$$

$$y=-\frac{14}{23}\approx -0.608695652$$

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.

$$3x-2\left(y+4\right)=0,4\left(x+1\right)+5y=10$$

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

$$3x-2\left(y+4\right)=0$$

Multiply -2 times y+4.

$$3x-2y-8=0$$

Add 8 to both sides of the equation.

$$3x-2y=8$$

Add 2y to both sides of the equation.

$$3x=2y+8$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(2y+8\right)$$

Multiply \frac{1}{3} times 8+2y.

$$x=\frac{2}{3}y+\frac{8}{3}$$

Substitute \frac{8+2y}{3} for x in the other equation, 4\left(x+1\right)+5y=10.

$$4\left(\frac{2}{3}y+\frac{8}{3}+1\right)+5y=10$$

Add \frac{8}{3} to 1.

$$4\left(\frac{2}{3}y+\frac{11}{3}\right)+5y=10$$

Multiply 4 times \frac{2y+11}{3}.

$$\frac{8}{3}y+\frac{44}{3}+5y=10$$

Add \frac{8y}{3} to 5y.

$$\frac{23}{3}y+\frac{44}{3}=10$$

Subtract \frac{44}{3} from both sides of the equation.

$$\frac{23}{3}y=-\frac{14}{3}$$

Divide both sides of the equation by \frac{23}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{14}{23}$$

Substitute -\frac{14}{23} for y in x=\frac{2}{3}y+\frac{8}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{2}{3}\left(-\frac{14}{23}\right)+\frac{8}{3}$$

Multiply \frac{2}{3} times -\frac{14}{23} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{28}{69}+\frac{8}{3}$$

Add \frac{8}{3} to -\frac{28}{69} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{52}{23}$$

The system is now solved.

$$x=\frac{52}{23},y=-\frac{14}{23}$$

Steps Using Matrices

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

$$3x-2\left(y+4\right)=0,4\left(x+1\right)+5y=10$$

Simplify the first equation to put it in standard form.

$$3x-2\left(y+4\right)=0$$

Multiply -2 times y+4.

$$3x-2y-8=0$$

Add 8 to both sides of the equation.

$$3x-2y=8$$

Simplify the second equation to put it in standard form.

$$4\left(x+1\right)+5y=10$$

Multiply 4 times x+1.

$$4x+4+5y=10$$

Subtract 4 from both sides of the equation.

$$4x+5y=6$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-2\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\6\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-2\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\4&5\end{matrix}\right))\left(\begin{matrix}3&-2\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&5\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&5\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&5\end{matrix}\right))\left(\begin{matrix}8\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3\times 5-\left(-2\times 4\right)}&-\frac{-2}{3\times 5-\left(-2\times 4\right)}\\-\frac{4}{3\times 5-\left(-2\times 4\right)}&\frac{3}{3\times 5-\left(-2\times 4\right)}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}&\frac{2}{23}\\-\frac{4}{23}&\frac{3}{23}\end{matrix}\right)\left(\begin{matrix}8\\6\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{23}\times 8+\frac{2}{23}\times 6\\-\frac{4}{23}\times 8+\frac{3}{23}\times 6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{52}{23}\\-\frac{14}{23}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{52}{23},y=-\frac{14}{23}$$