Question

Use the substitution method to solve the system 2.5x + y = -2, 3x + 2y = 0.

1. No solution
2. (-2, 3)
3. (3, -2)
4. Infinite solutions

Answer 1

To solve the system of equations using the substitution method, we have the following equations:

1. \( 2.5x + y = -2 \) (Equation 1)
2. \( 3x + 2y = 0 \) (Equation 2)

**Step 1: Solve one equation for one variable.**

Let's solve Equation 1 for \( y \):

\[
y = -2 - 2.5x
\]

**Step 2: Substitute into the second equation.**

Now, substitute \( y \) into Equation 2:

\[
3x + 2(-2 - 2.5x) = 0
\]

Distributing the 2:

\[
3x - 4 - 5x = 0
\]

Combine the \( x \) terms:

\[
-2x - 4 = 0
\]

**Step 3: Solve for \( x \).**

Add 4 to both sides:

\[
-2x = 4
\]

Divide by -2:

\[
x = -2
\]

**Step 4: Substitute back to find \( y \).**

Now substitute \( x = -2 \) back into the expression for \( y \):

\[
y = -2 - 2.5(-2)
\]
\[
y = -2 + 5
\]
\[
y = 3
\]

So, the solution to the system is:

\[
(x, y) = (-2, 3)
\]

**Conclusion:**

The solution to the system is \( (-2, 3) \).

So, the correct answer is **2. (-2, 3)**.