Question

use the elimination method to solve the system. \

$$\begin{cases}7a - 5b = 24 \\\\ 12a + 8b = 8\\end{cases}$$

(a, b) = (\boxed{})

Explanation:

Step1: Multiply equations to eliminate a variable

We have the system:

$$
LATEXBLOCK0
$$

To eliminate \(a\), we multiply equation \((1)\) by \(12\) and equation \((2)\) by \(7\):
Equation \((1)\times12\): \(84a - 60b = 288\)
Equation \((2)\times7\): \(84a + 56b = 56\)

Step2: Subtract the equations

Subtract the new equation \((2)\) from the new equation \((1)\):

$$
LATEXBLOCK1
$$

Step3: Solve for \(b\)

Divide both sides by \(-116\):

$$ b=\frac{232}{-116}=- 2 $$

Step4: Substitute \(b\) into equation \((1)\)

Substitute \(b = - 2\) into \(7a-5b = 24\):

$$
LATEXBLOCK2
$$

Step5: Solve for \(a\)

Divide both sides by \(7\):

$$ a=\frac{14}{7} = 2 $$

Answer:

\((2,-2)\)