Question

Use the substitution method to solve the system 9x−3y=12
9
𝑥
−
3
𝑦
=
12
, y+6=2x
𝑦
+
6
=
2
𝑥
.

Answer 1

To solve the system of equations using the substitution method, we first rearrange one of the equations to express one variable in terms of the other. Let's take the second equation:

1. **Equation 1:** \( 9x - 3y = 12 \)
2. **Equation 2:** \( y + 6 = 2x \)

We can rearrange the second equation to express \( y \) in terms of \( x \):

\[
y = 2x - 6
\]

Now that we have \( y \) in terms of \( x \), we can substitute this expression into the first equation:

\[
9x - 3(2x - 6) = 12
\]

Next, we simplify this equation:

\[
9x - 6x + 18 = 12
\]

Combining like terms:

\[
3x + 18 = 12
\]

Now, isolate \( x \) by subtracting 18 from both sides:

\[
3x = 12 - 18
\]

\[
3x = -6
\]

Now, divide by 3:

\[
x = -2
\]

Now that we have the value of \( x \), we can use it to find \( y \). Substitute \( x = -2 \) back into the equation \( y = 2x - 6 \):

\[
y = 2(-2) - 6
\]

\[
y = -4 - 6
\]

\[
y = -10
\]

Therefore, the solution to the system of equations is:

\[
\boxed{(-2, -10)}
\]

Finally, we can verify the answer by plugging \( x = -2 \) and \( y = -10 \) back into the original equations:

1. For \( 9x - 3y = 12 \):

\[
9(-2) - 3(-10) = -18 + 30 = 12 \, \text{(True)}
\]

2. For \( y + 6 = 2x \):

\[
-10 + 6 = 2(-2) \implies -4 = -4 \, \text{(True)}
\]

Both equations are satisfied, confirming that the solution is correct. Thus, the final answer is \( \boxed{(-2, -10)} \).