Question

Solve the system of equations below using elimination. You must show all work to earn full credit.



4x+6y=30


2x−6y=6

Answer 1

To solve the system of equations using the elimination method, we need to eliminate one variable by combining the two equations. The given equations are:

1) \( 4x + 6y = 30 \) (Equation 1)
2) \( 2x - 6y = 6 \) (Equation 2)

Let's first write down the equations:

**Step 1: Align the equations.**
Equation 1: \( 4x + 6y = 30 \)
Equation 2: \( 2x - 6y = 6 \)

**Step 2: Prepare to eliminate one of the variables.**
To eliminate \( y \), we can add the two equations directly since the coefficients of \( y \) are opposites (\(+6y\) and \(-6y\)).

**Step 3: Add the two equations.**
\[
(4x + 6y) + (2x - 6y) = 30 + 6
\]

Combining like terms gives us:
\[
4x + 2x + 6y - 6y = 30 + 6
\]
\[
6x = 36
\]

**Step 4: Solve for \( x \).**
Now we divide both sides by 6:
\[
x = \frac{36}{6} = 6
\]

**Step 5: Substitute \( x \) back into one of the original equations to solve for \( y \).**
We can substitute \( x = 6 \) into either Equation 1 or Equation 2. Let's use Equation 1:
\[
4x + 6y = 30
\]
Substituting \( x = 6 \):
\[
4(6) + 6y = 30
\]
\[
24 + 6y = 30
\]

**Step 6: Solve for \( y \).**
Now, subtract 24 from both sides:
\[
6y = 30 - 24
\]
\[
6y = 6
\]
Now divide both sides by 6:
\[
y = 1
\]

**Step 7: Present the solution.**
We have found \( x = 6 \) and \( y = 1 \).

The solution to the system of equations is:
\[
(x, y) = (6, 1)
\]

**Final Step: Verify the solution.**
Substituting \( x = 6 \) and \( y = 1 \) back into the original equations:

1. For Equation 1:
\[
4(6) + 6(1) = 24 + 6 = 30 \quad \text{(True)}
\]

2. For Equation 2:
\[
2(6) - 6(1) = 12 - 6 = 6 \quad \text{(True)}
\]

The solution satisfies both equations. Thus, the solution is:
\[
\boxed{(6, 1)}
\]