Question

After eliminating
in the first step by adding the following equations together, what equation will be left to solve for y
?
-6+5y=12
6x-9y=48

Answer 1

To eliminate one variable from the system of equations, we can manipulate the equations to add them together in such a way that one variable cancels out.

The given equations are:

1. \(-6 + 5y = 12\)
2. \(6x - 9y = 48\)

Let's solve the first equation for \(y\) in terms of \(x\) by isolating \(y\):

Starting with the first equation:
\[
-6 + 5y = 12
\]
Add 6 to both sides:
\[
5y = 12 + 6
\]
\[
5y = 18
\]
Now divide by 5:
\[
y = \frac{18}{5} = 3.6
\]

Now, we can plug that \(y\) back into the second equation:

Substituting \(y = 3.6\) into the second equation:
\[
6x - 9(3.6) = 48
\]
Calculate \(9 \times 3.6\):
\[
9 \times 3.6 = 32.4
\]
Now substitute back:
\[
6x - 32.4 = 48
\]
Add \(32.4\) to both sides:
\[
6x = 48 + 32.4
\]
\[
6x = 80.4
\]
Divide by \(6\):
\[
x = \frac{80.4}{6} = 13.4
\]

Now we have isolated both variables through elimination within substitutions. However, if your goal is simply to combine and eliminate \(x\) or \(y\) instead, you can manipulate equations without substitution.

In order to find the correct equation remaining after elimination and formation, we'll align with coefficient proportions.

Let’s go back and use the simplest elimination process directly to cancel \(y\) or \(x\).

1. Rearranging the first equation:
\[
5y = 12 + 6 \Rightarrow 5y = 18
\]

Thus, we may consider an equation with normalization:

Multiply the first equation by \(9\) to align \(y\):
\[
-54 + 45y = 108
\]
Combine it with second while using addition:
\[
6x - 9y = 48
\]

This doesn't yield a direct walk-through, but with coefficients:
Multiply the second by \(5\):
\[
30x - 45y = 240
\]

Adding:
\[
-54 + 30x = 108 + 240
\]
This leads to:
\[
30x - 45y = 240 \\
-54 + 30x = 348
\]

Equation becomes \(30x - 45y = 240\)

Thus, after manipulative efforts, you'll land with a linear track of equations' rearrangement leaning towards:

\[
0 = B(y) + C(x) - D
\]

Solving for either variable, we can express:
Ultimately, it's important to know that multiples can be shifted or manipulated as you want to reach a solution for either \(x\) or \(y\). **The descriptive equation that reveals the interaction of balances left:**
\[
28 - 9y = 80 or 45y -32.4 = 48
\]

This will ultimately guide you on \(y\), with equilibrium subject always on one or another variable based on elimination strategies.