91Ó°ÊÓ

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

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

Equation -

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

Write the equation in the standard form by Clear the fractions by multiplying the First equation by 3 and second equation by 6 -

$$\begin{gathered} (\frac{1}{3}x-y)×3=-3×3 \\ (\frac{2}{3}x+\frac{5}{2}y)×6=3×6 \\ x-3y=-9 \\ 4x+15y=18 \end{gathered}$$

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

$$\begin{gathered} (x-3y)×5=-9×5 \\ 4x+15y=18 \\ 5x-15y=-45 \\ 4x+15y=18 \end{gathered}$$

The equation become -

$$\begin{gathered} 5x-15y=-45 \\ 4x+15y=18 \end{gathered}$$

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

$\begin{array}{l}5x-15y=-45\\ \frac{4x+15y=18}{9x=-27}\end{array}$

Solve for x -

$x=-3$

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

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

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 -

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

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