Question

Solve a System of Equations by Elimination -

$$\begin{gathered} 6x-5y=-75 \\ -x-2y=-13 \end{gathered}$$

Short answer

The simplification of Expression will be - $(-5,9)$.

Step-by-step solution

Step 1.  Given Information.

Equation -$$\begin{gathered} 6x-5y=-75 \\ -x-2y=-13 \end{gathered}$$

Step 2.  To Find

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

Step 3.  Explanation

Equation -

$$\begin{gathered} 6x-5y=-75 \\ -x-2y=-13 \end{gathered}$$

We can eliminate y's by multiplying the first equation by 2 and second equation by 5 -

$$\begin{gathered} (6x-5y)\times 2=-75\times 2 \\ (-x-2y)\times 5=-13\times 5 \\ 12x-10y=-150 \\ -5x-10y=-65 \end{gathered}$$

Step 4.  Calculation

The equation become -

$$\begin{gathered} 12x-10y=-150 \\ -5x-10y=-65 \end{gathered}$$

We Subtractthe x's and y's and constants -

$\begin{array}{l}12x-10y=-150\\ \frac{-5x-10y=-65}{17x=-85}\end{array}$

Solve for x -

$x=-5$

Substitute $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{5}$into the second equation $\mathbf{-}\mathit{x}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{-}\mathbf{13}$, Then Solve for y -

$$\begin{gathered} -x-2y=-13 \\ -(-5)-2y=-13 \\ -2y=-13-5 \\ -2y=-18 \\ y=9 \end{gathered}$$

Written as$(x,y)$ will be $(-5,9)$

Step 5.  Check the answer

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

Put $(-5,9)$in the equations -

$$\begin{gathered} 6x-5y=-75 \\ 6\times (-5)-5\times 9=-75 \\ -30-45=-75 \\ -75=-75 \end{gathered}$$

$$\begin{gathered} -x-2y=-13 \\ -(-5)-2\times 9=-13 \\ 5-18=-13 \\ -13=-13 \end{gathered}$$

The Solution is -$(-5,9)$