Question

Use the substitution method to solve the system 2.5x+y=−2 , 3x+2y=0 .(1 point) Responses no solution no solution (−2,3) left parenthesis negative 2 comma 3 right parenthesis (3,−2) left parenthesis 3 comma negative 2 right parenthesis infinite solutions

Answer 1

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

We start with the two equations:

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

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

\[
y = -2 - 2.5x
\]

Now we will substitute this expression for \(y\) into Equation 2:

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

Now simplify:

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

Now that we have \(x = -2\), we can substitute it back into the equation we solved for \(y\):

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

So the solution to the system of equations is:

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

Thus, the correct answer is \((-2, 3)\).