Solve {l}{x+y=500}{24x+23y=118}

$$\left\{ \begin{array} { l } { x + y = 500 } \\ { 24 x + 23 y = 118 } \end{array} \right.$$

Solve for x, y

$$x=-11382$$

$$y=11882$$

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.

$$x+y=500,24x+23y=118$$

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

$$x+y=500$$

Subtract y from both sides of the equation.

$$x=-y+500$$

Substitute -y+500 for x in the other equation, 24x+23y=118.

$$24\left(-y+500\right)+23y=118$$

Multiply 24 times -y+500.

$$-24y+12000+23y=118$$

Add -24y to 23y.

$$-y+12000=118$$

Subtract 12000 from both sides of the equation.

$$-y=-11882$$

Divide both sides by -1.

$$y=11882$$

Substitute 11882 for y in x=-y+500. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-11882+500$$

Add 500 to -11882.

$$x=-11382$$

The system is now solved.

$$x=-11382,y=11882$$

Steps Using Matrices

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

$$x+y=500,24x+23y=118$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\24&23\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}500\\118\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\24&23\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\24&23\end{matrix}\right))\left(\begin{matrix}1&1\\24&23\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\24&23\end{matrix}\right))\left(\begin{matrix}500\\118\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\24&23\end{matrix}\right))\left(\begin{matrix}500\\118\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\24&23\end{matrix}\right))\left(\begin{matrix}500\\118\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{23-24}&-\frac{1}{23-24}\\-\frac{24}{23-24}&\frac{1}{23-24}\end{matrix}\right)\left(\begin{matrix}500\\118\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-23&1\\24&-1\end{matrix}\right)\left(\begin{matrix}500\\118\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-23\times 500+118\\24\times 500-118\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-11382\\11882\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-11382,y=11882$$

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.

$$x+y=500,24x+23y=118$$

To make x 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 1.

$$24x+24y=24\times 500,24x+23y=118$$

Simplify.

$$24x+24y=12000,24x+23y=118$$

Subtract 24x+23y=118 from 24x+24y=12000 by subtracting like terms on each side of the equal sign.

$$24x-24x+24y-23y=12000-118$$

Add 24x to -24x. Terms 24x and -24x cancel out, leaving an equation with only one variable that can be solved.

$$24y-23y=12000-118$$

Add 24y to -23y.

$$y=12000-118$$

Add 12000 to -118.

$$y=11882$$

Substitute 11882 for y in 24x+23y=118. Because the resulting equation contains only one variable, you can solve for x directly.

$$24x+23\times 11882=118$$

Multiply 23 times 11882.

$$24x+273286=118$$

Subtract 273286 from both sides of the equation.

$$24x=-273168$$

Divide both sides by 24.

$$x=-11382$$

The system is now solved.

$$x=-11382,y=11882$$