91Ó°ÊÓ

Equation -$$\begin{gathered} \frac{1}{3}x-y=-3 \\ x+\frac{5}{2}y=2 \end{gathered}$$

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

Equation -

$$\begin{gathered} \frac{1}{3}x-y=-3 \\ x+\frac{5}{2}y=2 \end{gathered}$$

Write the equation in the standard form by Clear the fractions by multiplying the First equation by 3 and second equation by 2 -

role="math" localid="1647779100912" $$\begin{gathered} \frac{1}{3}x-y=-3 \\ x+\frac{5}{2}y=2 \\ (\frac{1}{3}x-y)×3=-3×3 \\ (x+\frac{5}{2}y)×2=2×2 \\ x-3y=-9 \\ 2x+5y=4 \end{gathered}$$

We can eliminate y's by multiplying the first equation by 5 and second equation by 3 -

$$\begin{gathered} (x-3y)×5=-9×5 \\ (2x+5y)×3=4×3 \\ 5x-15y=-45 \\ 6x+15y=12 \end{gathered}$$

The equation become -

$$\begin{gathered} 5x-15y=-45 \\ 6x+15y=12 \end{gathered}$$

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

$\begin{array}{l}5x-15y=-45\\ \frac{6x+15y=12}{11x=-33}\end{array}$

Solve for x -

$x=-3$

Substitute $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{3}$into the second equation $\mathit{x}\mathbf{+}\frac{\mathbf{5}}{\mathbf{2}}\mathit{y}\mathbf{=}\mathbf{2}$, Then Solve for y -

$$\begin{gathered} x+\frac{5}{2}y=2 \\ -3+\frac{5}{2}y=2 \\ \frac{5}{2}y=2+3 \\ \frac{5}{2}y=5 \\ y=5×\frac{2}{5} \\ y=2 \end{gathered}$$

Written as $(x,y)$will be $(-3,2)$

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

Put $(-3,2)$in the equations -

$$\begin{gathered} \frac{1}{3}x-y=-3 \\ \frac{1}{3}×(-3)-2=-3 \\ -1-2=-3 \\ -3=-3 \\ x+\frac{5}{2}y=2 \\ -3+\frac{5}{2}×2=2 \\ -3+5=2 \\ 2=2 \end{gathered}$$

The Solution is - $\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{,}\mathbf{2}\mathbf{)}$