Solve {l}{5x+8y=120}{400x+500y=8200}

$$\left. \begin{array} { l } { 5 x + 8 y = 120 } \\ { 400 x + 500 y = 8200 } \end{array} \right.$$

Solve for x, y

$$x=8$$

$$y=10$$

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.

$$5x+8y=120,400x+500y=8200$$

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

$$5x+8y=120$$

Subtract 8y from both sides of the equation.

$$5x=-8y+120$$

Divide both sides by 5.

$$x=\frac{1}{5}\left(-8y+120\right)$$

Multiply \frac{1}{5} times -8y+120.

$$x=-\frac{8}{5}y+24$$

Substitute -\frac{8y}{5}+24 for x in the other equation, 400x+500y=8200.

$$400\left(-\frac{8}{5}y+24\right)+500y=8200$$

Multiply 400 times -\frac{8y}{5}+24.

$$-640y+9600+500y=8200$$

Add -640y to 500y.

$$-140y+9600=8200$$

Subtract 9600 from both sides of the equation.

$$-140y=-1400$$

Divide both sides by -140.

$$y=10$$

Substitute 10 for y in x=-\frac{8}{5}y+24. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{8}{5}\times 10+24$$

Multiply -\frac{8}{5} times 10.

$$x=-16+24$$

Add 24 to -16.

$$x=8$$

The system is now solved.

$$x=8,y=10$$

Steps Using Matrices

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

$$5x+8y=120,400x+500y=8200$$

Write the equations in matrix form.

$$\left(\begin{matrix}5&8\\400&500\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}120\\8200\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}5&8\\400&500\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&8\\400&500\end{matrix}\right))\left(\begin{matrix}5&8\\400&500\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&8\\400&500\end{matrix}\right))\left(\begin{matrix}120\\8200\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&8\\400&500\end{matrix}\right))\left(\begin{matrix}120\\8200\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&8\\400&500\end{matrix}\right))\left(\begin{matrix}120\\8200\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{500}{5\times 500-8\times 400}&-\frac{8}{5\times 500-8\times 400}\\-\frac{400}{5\times 500-8\times 400}&\frac{5}{5\times 500-8\times 400}\end{matrix}\right)\left(\begin{matrix}120\\8200\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{7}&\frac{2}{175}\\\frac{4}{7}&-\frac{1}{140}\end{matrix}\right)\left(\begin{matrix}120\\8200\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{7}\times 120+\frac{2}{175}\times 8200\\\frac{4}{7}\times 120-\frac{1}{140}\times 8200\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\10\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=8,y=10$$

Steps Using Elimination

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.

$$5x+8y=120,400x+500y=8200$$

To make 5x and 400x equal, multiply all terms on each side of the first equation by 400 and all terms on each side of the second by 5.

$$400\times 5x+400\times 8y=400\times 120,5\times 400x+5\times 500y=5\times 8200$$

Simplify.

$$2000x+3200y=48000,2000x+2500y=41000$$

Subtract 2000x+2500y=41000 from 2000x+3200y=48000 by subtracting like terms on each side of the equal sign.

$$2000x-2000x+3200y-2500y=48000-41000$$

Add 2000x to -2000x. Terms 2000x and -2000x cancel out, leaving an equation with only one variable that can be solved.

$$3200y-2500y=48000-41000$$

Add 3200y to -2500y.

$$700y=48000-41000$$

Add 48000 to -41000.

$$700y=7000$$

Divide both sides by 700.

$$y=10$$

Substitute 10 for y in 400x+500y=8200. Because the resulting equation contains only one variable, you can solve for x directly.

$$400x+500\times 10=8200$$

Multiply 500 times 10.

$$400x+5000=8200$$

Subtract 5000 from both sides of the equation.

$$400x=3200$$

Divide both sides by 400.

$$x=8$$

The system is now solved.

$$x=8,y=10$$