Question

Systems of Linear Equations Unit Test
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Question
Solve the system of equations.
6x−3y=−6

9x−3y=−18

Answer 1

To solve the system of equations

1. \( 6x - 3y = -6 \) (Equation 1)
2. \( 9x - 3y = -18 \) (Equation 2)

We can start by simplifying both equations.

### Step 1: Simplify each equation.

**For Equation 1:**
\[
6x - 3y = -6
\]
We can divide the entire equation by 3:
\[
2x - y = -2 \quad \text{(Equation 1 simplified)}
\]

**For Equation 2:**
\[
9x - 3y = -18
\]
We can divide the entire equation by 3:
\[
3x - y = -6 \quad \text{(Equation 2 simplified)}
\]

### Step 2: Subtract the simplified equations.

Now we can express \(y\) in terms of \(x\) from each equation.

From Equation 1 simplified:
\[
y = 2x + 2 \quad \text{(solving for y)}
\]

From Equation 2 simplified:
\[
y = 3x + 6 \quad \text{(solving for y)}
\]

### Step 3: Set the expressions for y equal to each other.

Since both equations equal \(y\), we can set them equal to find \(x\):
\[
2x + 2 = 3x + 6
\]

### Step 4: Solve for x.

Rearranging gives:
\[
2 - 6 = 3x - 2x \\
-4 = x
\]

### Step 5: Substitute \(x\) back into one of the equations to find y.

Using the equation \(y = 2x + 2\):
\[
y = 2(-4) + 2 \\
y = -8 + 2 \\
y = -6
\]

### Final Solution:

So, the solution to the system of equations is:
\[
(x, y) = (-4, -6)
\]

### Checking the Solution:

Substituting \(x = -4\) and \(y = -6\) back into the original equations:

1. For Equation 1:
\[
6(-4) - 3(-6) = -24 + 18 = -6 \quad \text{(True)}
\]

2. For Equation 2:
\[
9(-4) - 3(-6) = -36 + 18 = -18 \quad \text{(True)}
\]

Both equations are satisfied, confirming that the solution \((x, y) = (-4, -6)\) is correct.