Question

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

infinite solutions
infinite solutions

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

no solution
no solution

(3,−2)

Answer 1

To solve the system of equations using the substitution method, let's start with the given equations:

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

First, we can solve Equation 1 for \(y\):

\[
y = -2 - 2.5x
\]

Now that we have \(y\) in terms of \(x\), we can substitute this expression for \(y\) into Equation 2:

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

Now simplify and solve for \(x\):

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

Now that we have \(x\), we can substitute back into the equation we found for \(y\):

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

So, we find the solution to the system of equations is:

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

The system of equations has one unique solution at the point \((-2, 3)\).

The correct response is:

**(−2, 3)**