To solve the system of equations by elimination, the objective is to manipulate the given equations in such a way that one variable can be eliminated when the two equations are added or subtracted.
Given the system of equations:
[tex]\[
\left\{\begin{array}{l}
4x + y = 15 \\
3x - 2y = -8
\end{array}\right.
\][/tex]
Let's go through the steps:
1. Multiply the first equation by 2:
[tex]\[
2 \times (4x + y) = 2 \times 15
\][/tex]
This results in:
[tex]\[
8x + 2y = 30
\][/tex]
2. Multiply the second equation by 2:
[tex]\[
2 \times (3x - 2y) = 2 \times (-8)
\][/tex]
This results in:
[tex]\[
6x - 4y = -16
\][/tex]
3. Divide the second equation by 3:
[tex]\[
\frac{1}{3} \times (3x - 2y) = \frac{1}{3} \times (-8)
\][/tex]
This results in:
[tex]\[
x - \frac{2}{3}y = -\frac{8}{3}
\][/tex]
However, what we are primarily asked to do is to multiply the first equation by 2. This step is often taken to facilitate the elimination process by either adding or subtracting the resulting equations to eliminate one variable.
By multiplying the first equation by 2, we have:
[tex]\[
8x + 2y = 30
\][/tex]
This is a necessary step in the elimination method to combine it with another manipulated version of one of the original equations, aiming to eliminate one of the variables.
So, after multiplying the first equation by 2, your new resulting equation is:
[tex]\[
8x + 2y = 30.
\][/tex]
