Question

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

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

infinite solutions
infinite solutions

no solution
no solution

(−2,3)

Answer 1

To solve the system of equations using the substitution method, we start with the two equations:

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

**Step 1: Solve one of the equations for one variable.**

Let's solve the first equation for \( y \):

\[
y = -2 - 2.5x
\]

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

Now we substitute \( y \) in the second equation:

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

**Step 3: Simplify and solve for \( x \).**

Distributing the 2:

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

Combining like terms:

\[
-2x - 4 = 0
\]

Adding 4 to both sides:

\[
-2x = 4
\]

Dividing by -2:

\[
x = -2
\]

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

Plugging \( x = -2 \) into the equation for \( y \):

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

**Step 5: Write the solution as an ordered pair.**

The solution is \( (x, y) = (-2, 3) \).

**Final check: Substitute back into both original equations to verify.**

For the first equation:

\[
2.5(-2) + 3 = -5 + 3 = -2 \quad \text{(True)}
\]

For the second equation:

\[
3(-2) + 2(3) = -6 + 6 = 0 \quad \text{(True)}
\]

Since both equations are satisfied, the solution is:

\[
\boxed{(-2, 3)}
\]