Solve {l}{6x-6y-2y=0}{24x-48y=96}

$$\left\{ \begin{array} { l } { 6 x - 6 y - 2 y = 0 } \\ { 24 x - 48 y = 96 } \end{array} \right.$$

Solve for x, y

$$x=-8$$

$$y=-6$$

Steps Using Substitution

Consider the first equation. Combine -6y and -2y to get -8y.

$$6x-8y=0$$

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.

$$6x-8y=0,24x-48y=96$$

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

$$6x-8y=0$$

Add 8y to both sides of the equation.

$$6x=8y$$

Divide both sides by 6.

$$x=\frac{1}{6}\times 8y$$

Multiply \frac{1}{6} times 8y.

$$x=\frac{4}{3}y$$

Substitute \frac{4y}{3} for x in the other equation, 24x-48y=96.

$$24\times \frac{4}{3}y-48y=96$$

Multiply 24 times \frac{4y}{3}.

$$32y-48y=96$$

Add 32y to -48y.

$$-16y=96$$

Divide both sides by -16.

$$y=-6$$

Substitute -6 for y in x=\frac{4}{3}y. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{4}{3}\left(-6\right)$$

Multiply \frac{4}{3} times -6.

$$x=-8$$

The system is now solved.

$$x=-8,y=-6$$

Steps Using Matrices

Consider the first equation. Combine -6y and -2y to get -8y.

$$6x-8y=0$$

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

$$6x-8y=0,24x-48y=96$$

Write the equations in matrix form.

$$\left(\begin{matrix}6&-8\\24&-48\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\96\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}6&-8\\24&-48\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-8\\24&-48\end{matrix}\right))\left(\begin{matrix}6&-8\\24&-48\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-8\\24&-48\end{matrix}\right))\left(\begin{matrix}0\\96\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-8\\24&-48\end{matrix}\right))\left(\begin{matrix}0\\96\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-8\\24&-48\end{matrix}\right))\left(\begin{matrix}0\\96\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{48}{6\left(-48\right)-\left(-8\times 24\right)}&-\frac{-8}{6\left(-48\right)-\left(-8\times 24\right)}\\-\frac{24}{6\left(-48\right)-\left(-8\times 24\right)}&\frac{6}{6\left(-48\right)-\left(-8\times 24\right)}\end{matrix}\right)\left(\begin{matrix}0\\96\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{12}\\\frac{1}{4}&-\frac{1}{16}\end{matrix}\right)\left(\begin{matrix}0\\96\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}\times 96\\-\frac{1}{16}\times 96\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-8\\-6\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-8,y=-6$$

Steps Using Elimination

Consider the first equation. Combine -6y and -2y to get -8y.

$$6x-8y=0$$

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.

$$6x-8y=0,24x-48y=96$$

To make 6x 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 6.

$$24\times 6x+24\left(-8\right)y=0,6\times 24x+6\left(-48\right)y=6\times 96$$

Simplify.

$$144x-192y=0,144x-288y=576$$

Subtract 144x-288y=576 from 144x-192y=0 by subtracting like terms on each side of the equal sign.

$$144x-144x-192y+288y=-576$$

Add 144x to -144x. Terms 144x and -144x cancel out, leaving an equation with only one variable that can be solved.

$$-192y+288y=-576$$

Add -192y to 288y.

$$96y=-576$$

Divide both sides by 96.

$$y=-6$$

Substitute -6 for y in 24x-48y=96. Because the resulting equation contains only one variable, you can solve for x directly.

$$24x-48\left(-6\right)=96$$

Multiply -48 times -6.

$$24x+288=96$$

Subtract 288 from both sides of the equation.

$$24x=-192$$

Divide both sides by 24.

$$x=-8$$

The system is now solved.

$$x=-8,y=-6$$