Solve {l}{x+y=102}{12x+10y=118}

$$\left\{ \begin{array} { l } { x + y = 102 } \\ { 12 x + 10 y = 118 } \end{array} \right.$$

Solve for x, y

$$x=-451$$

$$y=553$$

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.

$$x+y=102,12x+10y=118$$

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

$$x+y=102$$

Subtract y from both sides of the equation.

$$x=-y+102$$

Substitute -y+102 for x in the other equation, 12x+10y=118.

$$12\left(-y+102\right)+10y=118$$

Multiply 12 times -y+102.

$$-12y+1224+10y=118$$

Add -12y to 10y.

$$-2y+1224=118$$

Subtract 1224 from both sides of the equation.

$$-2y=-1106$$

Divide both sides by -2.

$$y=553$$

Substitute 553 for y in x=-y+102. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-553+102$$

Add 102 to -553.

$$x=-451$$

The system is now solved.

$$x=-451,y=553$$

Steps Using Matrices

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

$$x+y=102,12x+10y=118$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\12&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}102\\118\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\12&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\12&10\end{matrix}\right))\left(\begin{matrix}1&1\\12&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\12&10\end{matrix}\right))\left(\begin{matrix}102\\118\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\12&10\end{matrix}\right))\left(\begin{matrix}102\\118\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\12&10\end{matrix}\right))\left(\begin{matrix}102\\118\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{10-12}&-\frac{1}{10-12}\\-\frac{12}{10-12}&\frac{1}{10-12}\end{matrix}\right)\left(\begin{matrix}102\\118\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5&\frac{1}{2}\\6&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}102\\118\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\times 102+\frac{1}{2}\times 118\\6\times 102-\frac{1}{2}\times 118\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-451\\553\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-451,y=553$$

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.

$$x+y=102,12x+10y=118$$

To make x and 12x equal, multiply all terms on each side of the first equation by 12 and all terms on each side of the second by 1.

$$12x+12y=12\times 102,12x+10y=118$$

Simplify.

$$12x+12y=1224,12x+10y=118$$

Subtract 12x+10y=118 from 12x+12y=1224 by subtracting like terms on each side of the equal sign.

$$12x-12x+12y-10y=1224-118$$

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

$$12y-10y=1224-118$$

Add 12y to -10y.

$$2y=1224-118$$

Add 1224 to -118.

$$2y=1106$$

Divide both sides by 2.

$$y=553$$

Substitute 553 for y in 12x+10y=118. Because the resulting equation contains only one variable, you can solve for x directly.

$$12x+10\times 553=118$$

Multiply 10 times 553.

$$12x+5530=118$$

Subtract 5530 from both sides of the equation.

$$12x=-5412$$

Divide both sides by 12.

$$x=-451$$

The system is now solved.

$$x=-451,y=553$$