Solve a System of Equations by Elimination -

$$\begin{gathered} 2x+9y=-4 \\ 3x+13y=-7 \end{gathered}$$

Equation - $$\begin{gathered} 2x+9y=-4 \\ 3x+13y=-7 \end{gathered}$$

Solve a System of Equations by Elimination and find the value of x and y.

Equation -

$$\begin{gathered} 2x+9y=-4 \\ 3x+13y=-7 \end{gathered}$$

We can eliminate y's by multiplying the first equation by 13 and second equation by 9 -

$$\begin{gathered} (2x+9y)\times 13=-4\times 13 \\ (3x+13y)\times 9=-7\times 9 \\ 26x+117y=-52 \\ 27x+117y=-63 \end{gathered}$$

The equation become -

$$\begin{gathered} 26x+117y=-52 \\ 27x+117y=-63 \end{gathered}$$

We subtract the x's and y's and constants -

$\begin{array}{l}26x+117y=-52\\ \frac{27x+117y=-63}{-x=11}\end{array}$

Solve for x -

$x=-11$

Substitute $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{11}$into the second equation$\mathbf{3}\mathit{x}\mathbf{+}\mathbf{13}\mathit{y}\mathbf{=}\mathbf{-}\mathbf{7}$, Then Solve for y -

$$\begin{gathered} 3x+13y=-7 \\ 3\times -11+13y=-7 \\ -33+13y=-7 \\ 13y=-7+33 \\ 13y=26 \\ y=2 \end{gathered}$$

Written as $(x,y)$will be $(-11,2)$

Check that the ordered pair is a solution to both original equations -

Put $(-11,2)$in the equations -

$$\begin{gathered} 2x+9y=-4 \\ 2\times (-11)+9\times 2=-4 \\ -22+18=-4 \\ -4=-4 \\ 3x+13y=-7 \\ 3\times (-11)+13\times 2=-7 \\ -33+26=-7 \\ -7=-7 \end{gathered}$$

The Solution is $\mathbf{(}\mathbf{-}\mathbf{11}\mathbf{,}\mathbf{2}\mathbf{)}$