Solve Systems of Equations by Elimination: example 5.27

Solve the system by elimination. ${\begin{matrix} 3 x - 2 y = −2 \\ 5 x - 6 y = 10 \end{matrix}$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solve the system by elimination. ${\begin{matrix} 4 x - 3 y = 1 \\ 5 x - 9 y = −4 \end{matrix}$

How did it go?


Both equations are in standard form.
None of the coefficients are opposites.
We can make the coefficients of y opposites by multiplying the first equation by −3.
Simplify.
Add the two equations to eliminate y.
Solve for the remaining variable, x. Substitute x = −4 into one of the original equations.

Solve for y.
Write the solution as an ordered pair.	The ordered pair is (−4, −5).
Check that the ordered pair is a solution to both original equations. $\begin{matrix} \begin{matrix} 3 x - 2 y & = & - 2 \\ 3 (- 4) - 2 (- 5) & \overset{?}{=} & - 2 \\ - 12 + 10 & \overset{?}{=} & - 2 \\ - 2 y & = & - 2 ✓ \end{matrix} & \begin{matrix} 5 x - 6 y & = & 10 \\ 3 (- 4) - 6 (- 5) & \overset{?}{=} & 10 \\ - 20 + 30 & \overset{?}{=} & 10 \\ 10 & = & 10 ✓ \end{matrix} \end{matrix}$
	The solution is (−4, −5).


Both equations are in standard form.
None of the coefficients are opposites.
We can make the coefficients of y opposites by multiplying the first equation by −3.
Simplify.
Add the two equations to eliminate y.
Solve for the remaining variable, x. Substitute x = −4 into one of the original equations.

Solve for y.
Write the solution as an ordered pair.	The ordered pair is (−4, −5).
Check that the ordered pair is a solution to both original equations. $\begin{matrix} \begin{matrix} 3 x - 2 y & = & - 2 \\ 3 (- 4) - 2 (- 5) & \overset{?}{=} & - 2 \\ - 12 + 10 & \overset{?}{=} & - 2 \\ - 2 y & = & - 2 ✓ \end{matrix} & \begin{matrix} 5 x - 6 y & = & 10 \\ 3 (- 4) - 6 (- 5) & \overset{?}{=} & 10 \\ - 20 + 30 & \overset{?}{=} & 10 \\ 10 & = & 10 ✓ \end{matrix} \end{matrix}$
	The solution is (−4, −5).