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 no solution no solution infinite solutions infinite solutions (−2,3) left parenthesis negative 2 comma 3 right parenthesis

Answer 1

To solve the system of equations using the substitution method, we start with the two 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 this expression for \( y \) into the second equation.**

Substituting \( y \) in Equation 2:

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

Now simplify this equation:

\[
3x - 4 - 5x = 0
\]
\[
-2x - 4 = 0
\]
\[
-2x = 4
\]
\[
x = -2
\]

**Step 3: Substitute \( x \) back into the equation for \( y \).**

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

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

**Step 4: Write the solution.**

Thus, the solution to the system is:

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

So the answer is \( (-2, 3) \).