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:
\[

$$\begin{cases} 7a - 5b = 24 \quad (1)\\ 12a + 8b = 8 \quad (2) \end{cases}$$

\]
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 \((1)\):
\[

$$\begin{align*} (84a - 60b)-(84a + 56b)&=288 - 56\\ 84a - 60b - 84a - 56b&=232\\ - 116b&=232 \end{align*}$$

\]

Step3: Solve for \(b\)

Divide both sides by \(-116\):
\[
b=\frac{232}{-116}=- 2
\]

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

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

$$\begin{align*} 7a-5\times(-2)&=24\\ 7a + 10&=24\\ 7a&=24 - 10\\ 7a&=14 \end{align*}$$

\]

Step5: Solve for \(a\)

Divide both sides by \(7\):
\[
a=\frac{14}{7} = 2
\]

Answer:

\((2,-2)\)