Solve the system using elimination. 5x +3y=-6 and 3x-2y=4

To solve the system of equations using the elimination method, we start with the given equations:

\[
5x + 3y = -6 \quad \text{(Equation 1)}
\]
\[
3x - 2y = 4 \quad \text{(Equation 2)}
\]

### Step 1: Make the coefficients of one variable the same
We'll eliminate \( x \) by making the coefficients of \( x \) the same in both equations. We can do this by multiplying both equations by suitable numbers:

- Multiply Equation 1 by 3:
  \[
  3(5x + 3y) = 3(-6)
  \]
  \[
  15x + 9y = -18 \quad \text{(Equation 3)}
  \]

- Multiply Equation 2 by 5:
  \[
  5(3x - 2y) = 5(4)
  \]
  \[
  15x - 10y = 20 \quad \text{(Equation 4)}
  \]

### Step 2: Subtract the equations to eliminate \( x \)
Now, subtract Equation 4 from Equation 3:

\[
(15x + 9y) - (15x - 10y) = -18 - 20
\]

Simplifying, we get:

\[
15x + 9y - 15x + 10y = -38
\]

\[
19y = -38
\]

### Step 3: Solve for \( y \)
Divide both sides by 19:

\[
y = \frac{-38}{19}
\]

\[
y = -2
\]

### Step 4: Substitute \( y \) back into one of the original equations to find \( x \)
We'll substitute \( y = -2 \) into Equation 1:

\[
5x + 3(-2) = -6
\]

Simplify:

\[
5x - 6 = -6
\]

Add 6 to both sides:

\[
5x = 0
\]

Divide by 5:

\[
x = 0
\]

### Solution
The solution to the system of equations is \( x = 0 \) and \( y = -2 \).

Would you like further details or have any questions? Here are some related questions you might consider:

1. How does the elimination method differ from the substitution method?
2. What are the advantages of using the elimination method?
3. How can you solve a system of equations using matrices?
4. What if both variables were eliminated during the elimination process?
5. How do you determine if a system of equations has no solution or infinitely many solutions?

**Tip:** When solving systems of equations, always double-check your steps by substituting your solution back into the original equations.

Learn how to solve the system of equations 5x + 3y = -6 and 3x - 2y = 4 using the elimination method. This problem covers key concepts such as systems of linear equations and the elimination method. Suitable for students in Grades 9-12.

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