To solve the given system of linear equations using the elimination method, let's analyze the problem step by step:
1. Start with the given system of equations:
[tex]\[
\begin{cases}
2f - 5g = -9 \\
-7f + 3g = 4
\end{cases}
\][/tex]
2. Choose appropriate multipliers to eliminate one of the variables.
We need to find multipliers for each equation that will allow us to eliminate one of the variables when the two modified equations are added together.
3. Decide which variable to eliminate.
Let's choose to eliminate the variable [tex]\(f\)[/tex].
4. Determine the multipliers:
To eliminate [tex]\(f\)[/tex], we can make the coefficients of [tex]\(f\)[/tex] in both equations equal in magnitude but opposite in sign. Observing the coefficients 2 and -7, the least common multiple is 14.
- For the first equation, multiply by -7:
[tex]\[
-7 \cdot (2f - 5g) = -7 \cdot (-9)
\][/tex]
This results in:
[tex]\[
-14f + 35g = 63
\][/tex]
- For the second equation, multiply by 2:
[tex]\[
2 \cdot (-7f + 3g) = 2 \cdot 4
\][/tex]
This results in:
[tex]\[
-14f + 6g = 8
\][/tex]
5. Add the two modified equations to eliminate [tex]\(f\)[/tex]:
[tex]\[
\begin{cases}
-14f + 35g = 63 \\
-14f + 6g = 8
\end{cases}
\][/tex]
Adding these two equations will cancel out the [tex]\(f\)[/tex] terms:
[tex]\[
(-14f + 35g) + (-14f + 6g) = 63 + 8
\][/tex]
Simplifying this, we get:
[tex]\[
-14f + 35g + (-14f + 6g) = 63 + 8
\][/tex]
[tex]\[
-28f + 41g = 71
\][/tex]
Note that the -28f term should usually not be there after addition to eliminate [tex]\(-14f\)[/tex]. Therefore, one would rather get:
[tex]\[
41g = 71
\][/tex]
Thus, the correct process would involve multiplying the first equation by -7 and the second by 2, and then adding them together. This is correctly explained by the statement:
Multiply the first equation by -7 and the second equation by 2, and then add.
