Solve {l}{x+11y=154}{26x-y=273}

$$\left. \begin{array} { l } { x + 11 y = 154 } \\ { 26 x - y = 273 } \end{array} \right.$$

Solve for x, y

$$x=11$$

$$y=13$$

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.

$$x+11y=154,26x-y=273$$

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

$$x+11y=154$$

Subtract 11y from both sides of the equation.

$$x=-11y+154$$

Substitute -11y+154 for x in the other equation, 26x-y=273.

$$26\left(-11y+154\right)-y=273$$

Multiply 26 times -11y+154.

$$-286y+4004-y=273$$

Add -286y to -y.

$$-287y+4004=273$$

Subtract 4004 from both sides of the equation.

$$-287y=-3731$$

Divide both sides by -287.

$$y=13$$

Substitute 13 for y in x=-11y+154. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-11\times 13+154$$

Multiply -11 times 13.

$$x=-143+154$$

Add 154 to -143.

$$x=11$$

The system is now solved.

$$x=11,y=13$$

Steps Using Matrices

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

$$x+11y=154,26x-y=273$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&11\\26&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}154\\273\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&11\\26&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&11\\26&-1\end{matrix}\right))\left(\begin{matrix}1&11\\26&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&11\\26&-1\end{matrix}\right))\left(\begin{matrix}154\\273\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&11\\26&-1\end{matrix}\right))\left(\begin{matrix}154\\273\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&11\\26&-1\end{matrix}\right))\left(\begin{matrix}154\\273\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-11\times 26}&-\frac{11}{-1-11\times 26}\\-\frac{26}{-1-11\times 26}&\frac{1}{-1-11\times 26}\end{matrix}\right)\left(\begin{matrix}154\\273\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{287}&\frac{11}{287}\\\frac{26}{287}&-\frac{1}{287}\end{matrix}\right)\left(\begin{matrix}154\\273\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{287}\times 154+\frac{11}{287}\times 273\\\frac{26}{287}\times 154-\frac{1}{287}\times 273\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\13\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=11,y=13$$

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.

$$x+11y=154,26x-y=273$$

To make x and 26x equal, multiply all terms on each side of the first equation by 26 and all terms on each side of the second by 1.

$$26x+26\times 11y=26\times 154,26x-y=273$$

Simplify.

$$26x+286y=4004,26x-y=273$$

Subtract 26x-y=273 from 26x+286y=4004 by subtracting like terms on each side of the equal sign.

$$26x-26x+286y+y=4004-273$$

Add 26x to -26x. Terms 26x and -26x cancel out, leaving an equation with only one variable that can be solved.

$$286y+y=4004-273$$

Add 286y to y.

$$287y=4004-273$$

Add 4004 to -273.

$$287y=3731$$

Divide both sides by 287.

$$y=13$$

Substitute 13 for y in 26x-y=273. Because the resulting equation contains only one variable, you can solve for x directly.

$$26x-13=273$$

Add 13 to both sides of the equation.

$$26x=286$$

Divide both sides by 26.

$$x=11$$

The system is now solved.

$$x=11,y=13$$