Question

Use the substitution method to solve the system 2.5x + y = −2, 3x + 2y = 0 .(1 point)
Responses

infinite solutions
infinite solutions

(3,−2)
left parenthesis 3 comma negative 2 right parenthesis

no solution
no solution

(−2,3)

Answer 1

To solve the system of equations using the substitution method, we first solve one of the equations for one variable and then substitute that expression into the other equation.

Given the system:
1. \( 2.5x + y = -2 \) (Equation 1)
2. \( 3x + 2y = 0 \) (Equation 2)

### Step 1: Solve Equation 1 for \( y \)

From Equation 1:

\[
y = -2 - 2.5x
\]

### Step 2: Substitute \( y \) in Equation 2

Now, we substitute \( y \) in Equation 2:

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

Simplifying the equation:

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

\[
-2x - 4 = 0
\]

### Step 3: Solve for \( x \)

\[
-2x = 4
\]

\[
x = -2
\]

### Step 4: Substitute \( x \) back to find \( y \)

Now substitute \( x = -2 \) back into the equation we solved for \( y \):

\[
y = -2 - 2.5(-2)
\]

\[
y = -2 + 5 = 3
\]

### Solution

The solution to the system of equations is:

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

### Final Response

The correct response among the options is:

**(−2,3)**