Question

use elimination to solve the system of equations.
$-3x - 8y = -24$
$3x - 5y = 4.5$
(\boxed{\quad}, \boxed{\quad})$

Explanation:

Step1: Add the two equations to eliminate \(x\)

We have the system:
\[

$$\begin{cases} -3x - 8y = -24 \\ 3x - 5y = 4.5 \end{cases}$$

\]
Adding the left - hand sides and the right - hand sides of the two equations:
\((-3x - 8y)+(3x - 5y)=-24 + 4.5\)
Simplify the left - hand side: \(-3x+3x-8y - 5y=-13y\)
Simplify the right - hand side: \(-24 + 4.5=-19.5\)
So we get the equation \(-13y=-19.5\)

Step2: Solve for \(y\)

To solve for \(y\), divide both sides of the equation \(-13y=-19.5\) by \(-13\):
\(y=\frac{-19.5}{-13}\)
\(y = 1.5\)

Step3: Substitute \(y = 1.5\) into one of the original equations to solve for \(x\)

Let's substitute \(y = 1.5\) into the second equation \(3x-5y = 4.5\)
We have \(3x-5\times1.5=4.5\)
First, calculate \(5\times1.5 = 7.5\), so the equation becomes \(3x-7.5 = 4.5\)
Add \(7.5\) to both sides of the equation: \(3x=4.5 + 7.5\)
\(3x=12\)
Divide both sides by \(3\): \(x=\frac{12}{3}=4\)

Answer:

\((4,1.5)\)