Solve {l}{x-2-3y=9}{x-2/6-2y=3}

$$\left\{ \begin{array} { l } { x - 2 - 3 y = 9 } \\ { \frac { x - 2 } { 6 } - 2 y = 3 } \end{array} \right.$$

Solve for x, y

$$x=8$$

$$y=-1$$

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.

$$x-3y-2=9,\frac{1}{6}\left(x-2\right)-2y=3$$

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

$$x-3y-2=9$$

Add 2 to both sides of the equation.

$$x-3y=11$$

Add 3y to both sides of the equation.

$$x=3y+11$$

Substitute 3y+11 for x in the other equation, \frac{1}{6}\left(x-2\right)-2y=3.

$$\frac{1}{6}\left(3y+11-2\right)-2y=3$$

Add 11 to -2.

$$\frac{1}{6}\left(3y+9\right)-2y=3$$

Multiply \frac{1}{6} times 9+3y.

$$\frac{1}{2}y+\frac{3}{2}-2y=3$$

Add \frac{y}{2} to -2y.

$$-\frac{3}{2}y+\frac{3}{2}=3$$

Subtract \frac{3}{2} from both sides of the equation.

$$-\frac{3}{2}y=\frac{3}{2}$$

Divide both sides of the equation by -\frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-1$$

Substitute -1 for y in x=3y+11. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=3\left(-1\right)+11$$

Multiply 3 times -1.

$$x=-3+11$$

Add 11 to -3.

$$x=8$$

The system is now solved.

$$x=8,y=-1$$

Steps Using Matrices

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

$$x-3y-2=9,\frac{1}{6}\left(x-2\right)-2y=3$$

Simplify the second equation to put it in standard form.

$$\frac{1}{6}\left(x-2\right)-2y=3$$

Multiply \frac{1}{6} times x-2.

$$\frac{1}{6}x-\frac{1}{3}-2y=3$$

Add \frac{1}{3} to both sides of the equation.

$$\frac{1}{6}x-2y=\frac{10}{3}$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-3\\\frac{1}{6}&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\\frac{10}{3}\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-3\\\frac{1}{6}&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\\frac{1}{6}&-2\end{matrix}\right))\left(\begin{matrix}1&-3\\\frac{1}{6}&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\\frac{1}{6}&-2\end{matrix}\right))\left(\begin{matrix}11\\\frac{10}{3}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\\frac{1}{6}&-2\end{matrix}\right))\left(\begin{matrix}11\\\frac{10}{3}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\\frac{1}{6}&-2\end{matrix}\right))\left(\begin{matrix}11\\\frac{10}{3}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-\left(-3\times \frac{1}{6}\right)}&-\frac{-3}{-2-\left(-3\times \frac{1}{6}\right)}\\-\frac{\frac{1}{6}}{-2-\left(-3\times \frac{1}{6}\right)}&\frac{1}{-2-\left(-3\times \frac{1}{6}\right)}\end{matrix}\right)\left(\begin{matrix}11\\\frac{10}{3}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}&-2\\\frac{1}{9}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}11\\\frac{10}{3}\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}\times 11-2\times \frac{10}{3}\\\frac{1}{9}\times 11-\frac{2}{3}\times \frac{10}{3}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\-1\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=8,y=-1$$