Solve {l}{2x+y=45}{12(2(x+5))=y}

$$\left\{ \begin{array} { l } { 2 x + y = 45 } \\ { 12 ( 2 ( x + 5 ) ) = y } \end{array} \right.$$

Solve for x, y

$$x = -\frac{75}{26} = -2\frac{23}{26} \approx -2.884615385$$

$$y = \frac{660}{13} = 50\frac{10}{13} \approx 50.769230769$$

Steps Using Substitution

Consider the second equation. Multiply 12 and 2 to get 24.

$$24\left(x+5\right)=y$$

Use the distributive property to multiply 24 by x+5.

$$24x+120=y$$

Subtract y from both sides.

$$24x+120-y=0$$

Subtract 120 from both sides. Anything subtracted from zero gives its negation.

$$24x-y=-120$$

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.

$$2x+y=45,24x-y=-120$$

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

$$2x+y=45$$

Subtract y from both sides of the equation.

$$2x=-y+45$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-y+45\right)$$

Multiply \frac{1}{2} times -y+45.

$$x=-\frac{1}{2}y+\frac{45}{2}$$

Substitute \frac{-y+45}{2} for x in the other equation, 24x-y=-120.

$$24\left(-\frac{1}{2}y+\frac{45}{2}\right)-y=-120$$

Multiply 24 times \frac{-y+45}{2}.

$$-12y+540-y=-120$$

Add -12y to -y.

$$-13y+540=-120$$

Subtract 540 from both sides of the equation.

$$-13y=-660$$

Divide both sides by -13.

$$y=\frac{660}{13}$$

Substitute \frac{660}{13} for y in x=-\frac{1}{2}y+\frac{45}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{2}\times \frac{660}{13}+\frac{45}{2}$$

Multiply -\frac{1}{2} times \frac{660}{13} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{330}{13}+\frac{45}{2}$$

Add \frac{45}{2} to -\frac{330}{13} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{75}{26}$$

The system is now solved.

$$x=-\frac{75}{26},y=\frac{660}{13}$$

Steps Using Matrices

Consider the second equation. Multiply 12 and 2 to get 24.

$$24\left(x+5\right)=y$$

Use the distributive property to multiply 24 by x+5.

$$24x+120=y$$

Subtract y from both sides.

$$24x+120-y=0$$

Subtract 120 from both sides. Anything subtracted from zero gives its negation.

$$24x-y=-120$$

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

$$2x+y=45,24x-y=-120$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&1\\24&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\-120\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&1\\24&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\24&-1\end{matrix}\right))\left(\begin{matrix}2&1\\24&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\24&-1\end{matrix}\right))\left(\begin{matrix}45\\-120\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\24&-1\end{matrix}\right))\left(\begin{matrix}45\\-120\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\24&-1\end{matrix}\right))\left(\begin{matrix}45\\-120\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-24}&-\frac{1}{2\left(-1\right)-24}\\-\frac{24}{2\left(-1\right)-24}&\frac{2}{2\left(-1\right)-24}\end{matrix}\right)\left(\begin{matrix}45\\-120\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{26}&\frac{1}{26}\\\frac{12}{13}&-\frac{1}{13}\end{matrix}\right)\left(\begin{matrix}45\\-120\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{26}\times 45+\frac{1}{26}\left(-120\right)\\\frac{12}{13}\times 45-\frac{1}{13}\left(-120\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{75}{26}\\\frac{660}{13}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{75}{26},y=\frac{660}{13}$$

Steps Using Elimination

Consider the second equation. Multiply 12 and 2 to get 24.

$$24\left(x+5\right)=y$$

Use the distributive property to multiply 24 by x+5.

$$24x+120=y$$

Subtract y from both sides.

$$24x+120-y=0$$

Subtract 120 from both sides. Anything subtracted from zero gives its negation.

$$24x-y=-120$$

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.

$$2x+y=45,24x-y=-120$$

To make 2x and 24x equal, multiply all terms on each side of the first equation by 24 and all terms on each side of the second by 2.

$$24\times 2x+24y=24\times 45,2\times 24x+2\left(-1\right)y=2\left(-120\right)$$

Simplify.

$$48x+24y=1080,48x-2y=-240$$

Subtract 48x-2y=-240 from 48x+24y=1080 by subtracting like terms on each side of the equal sign.

$$48x-48x+24y+2y=1080+240$$

Add 48x to -48x. Terms 48x and -48x cancel out, leaving an equation with only one variable that can be solved.

$$24y+2y=1080+240$$

Add 24y to 2y.

$$26y=1080+240$$

Add 1080 to 240.

$$26y=1320$$

Divide both sides by 26.

$$y=\frac{660}{13}$$

Substitute \frac{660}{13} for y in 24x-y=-120. Because the resulting equation contains only one variable, you can solve for x directly.

$$24x-\frac{660}{13}=-120$$

Add \frac{660}{13} to both sides of the equation.

$$24x=-\frac{900}{13}$$

Divide both sides by 24.

$$x=-\frac{75}{26}$$

The system is now solved.

$$x=-\frac{75}{26},y=\frac{660}{13}$$