Solve {l}{3y-6x=0}{19x+17y=0}

$$\left. \begin{array} { l } { 3 y - 6 x = 0 } \\ { 19 x + 17 y = 0 } \end{array} \right.$$

Solve for y, x

$$x=0$$

$$y=0$$

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.

$$3y-6x=0,17y+19x=0$$

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

$$3y-6x=0$$

Add 6x to both sides of the equation.

$$3y=6x$$

Divide both sides by 3.

$$y=\frac{1}{3}\times 6x$$

Multiply \frac{1}{3} times 6x.

$$y=2x$$

Substitute 2x for y in the other equation, 17y+19x=0.

$$17\times 2x+19x=0$$

Multiply 17 times 2x.

$$34x+19x=0$$

Add 34x to 19x.

$$53x=0$$

Divide both sides by 53.

$$x=0$$

Substitute 0 for x in y=2x. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=0$$

The system is now solved.

$$y=0,x=0$$

Steps Using Matrices

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

$$3y-6x=0,17y+19x=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-6\\17&19\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-6\\17&19\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-6\\17&19\end{matrix}\right))\left(\begin{matrix}3&-6\\17&19\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\17&19\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\17&19\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-6\\17&19\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{19}{3\times 19-\left(-6\times 17\right)}&-\frac{-6}{3\times 19-\left(-6\times 17\right)}\\-\frac{17}{3\times 19-\left(-6\times 17\right)}&\frac{3}{3\times 19-\left(-6\times 17\right)}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{19}{159}&\frac{2}{53}\\-\frac{17}{159}&\frac{1}{53}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=0,x=0$$

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.

$$3y-6x=0,17y+19x=0$$

To make 3y and 17y equal, multiply all terms on each side of the first equation by 17 and all terms on each side of the second by 3.

$$17\times 3y+17\left(-6\right)x=0,3\times 17y+3\times 19x=0$$

Simplify.

$$51y-102x=0,51y+57x=0$$

Subtract 51y+57x=0 from 51y-102x=0 by subtracting like terms on each side of the equal sign.

$$51y-51y-102x-57x=0$$

Add 51y to -51y. Terms 51y and -51y cancel out, leaving an equation with only one variable that can be solved.

$$-102x-57x=0$$

Add -102x to -57x.

$$-159x=0$$

Divide both sides by -159.

$$x=0$$

Substitute 0 for x in 17y+19x=0. Because the resulting equation contains only one variable, you can solve for y directly.

$$17y=0$$

Divide both sides by 17.

$$y=0$$

The system is now solved.

$$y=0,x=0$$