Solve {l}{x-1/6+y=6}{y-1/4+x=8}

$$\left. \begin{array} { l } { \frac { x - 1 } { 6 } + y = 6 } \\ { \frac { y - 1 } { 4 } + x = 8 } \end{array} \right.$$

Solve for x, y

$$x=7$$

$$y=5$$

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.

$$\frac{1}{6}\left(x-1\right)+y=6,x+\frac{1}{4}\left(y-1\right)=8$$

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

$$\frac{1}{6}\left(x-1\right)+y=6$$

Multiply \frac{1}{6} times x-1.

$$\frac{1}{6}x-\frac{1}{6}+y=6$$

Add \frac{1}{6} to both sides of the equation.

$$\frac{1}{6}x+y=\frac{37}{6}$$

Subtract y from both sides of the equation.

$$\frac{1}{6}x=-y+\frac{37}{6}$$

Multiply both sides by 6.

$$x=6\left(-y+\frac{37}{6}\right)$$

Multiply 6 times -y+\frac{37}{6}.

$$x=-6y+37$$

Substitute -6y+37 for x in the other equation, x+\frac{1}{4}\left(y-1\right)=8.

$$-6y+37+\frac{1}{4}\left(y-1\right)=8$$

Multiply \frac{1}{4} times y-1.

$$-6y+37+\frac{1}{4}y-\frac{1}{4}=8$$

Add -6y to \frac{y}{4}.

$$-\frac{23}{4}y+37-\frac{1}{4}=8$$

Add 37 to -\frac{1}{4}.

$$-\frac{23}{4}y+\frac{147}{4}=8$$

Subtract \frac{147}{4} from both sides of the equation.

$$-\frac{23}{4}y=-\frac{115}{4}$$

Divide both sides of the equation by -\frac{23}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=5$$

Substitute 5 for y in x=-6y+37. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-6\times 5+37$$

Multiply -6 times 5.

$$x=-30+37$$

Add 37 to -30.

$$x=7$$

The system is now solved.

$$x=7,y=5$$

Steps Using Matrices

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

$$\frac{1}{6}\left(x-1\right)+y=6,x+\frac{1}{4}\left(y-1\right)=8$$

Simplify the first equation to put it in standard form.

$$\frac{1}{6}\left(x-1\right)+y=6$$

Multiply \frac{1}{6} times x-1.

$$\frac{1}{6}x-\frac{1}{6}+y=6$$

Add \frac{1}{6} to both sides of the equation.

$$\frac{1}{6}x+y=\frac{37}{6}$$

Simplify the second equation to put it in standard form.

$$x+\frac{1}{4}\left(y-1\right)=8$$

Multiply \frac{1}{4} times y-1.

$$x+\frac{1}{4}y-\frac{1}{4}=8$$

Add \frac{1}{4} to both sides of the equation.

$$x+\frac{1}{4}y=\frac{33}{4}$$

Write the equations in matrix form.

$$\left(\begin{matrix}\frac{1}{6}&1\\1&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{37}{6}\\\frac{33}{4}\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}\frac{1}{6}&1\\1&\frac{1}{4}\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{6}&1\\1&\frac{1}{4}\end{matrix}\right))\left(\begin{matrix}\frac{1}{6}&1\\1&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{6}&1\\1&\frac{1}{4}\end{matrix}\right))\left(\begin{matrix}\frac{37}{6}\\\frac{33}{4}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{6}&1\\1&\frac{1}{4}\end{matrix}\right))\left(\begin{matrix}\frac{37}{6}\\\frac{33}{4}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{6}&1\\1&\frac{1}{4}\end{matrix}\right))\left(\begin{matrix}\frac{37}{6}\\\frac{33}{4}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{4}}{\frac{1}{6}\times \frac{1}{4}-1}&-\frac{1}{\frac{1}{6}\times \frac{1}{4}-1}\\-\frac{1}{\frac{1}{6}\times \frac{1}{4}-1}&\frac{\frac{1}{6}}{\frac{1}{6}\times \frac{1}{4}-1}\end{matrix}\right)\left(\begin{matrix}\frac{37}{6}\\\frac{33}{4}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{23}&\frac{24}{23}\\\frac{24}{23}&-\frac{4}{23}\end{matrix}\right)\left(\begin{matrix}\frac{37}{6}\\\frac{33}{4}\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{23}\times \frac{37}{6}+\frac{24}{23}\times \frac{33}{4}\\\frac{24}{23}\times \frac{37}{6}-\frac{4}{23}\times \frac{33}{4}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\5\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=7,y=5$$