Question

Use the substitution method to solve the system 2x+3y=4, −4x−6y=−8.(1 point)
Responses

(−1,2)
left parenthesis negative 1 comma 2 right parenthesis

no solution
no solution

(2−1)
left parenthesis 2 comma negative 1 right parenthesis

infinite solutions

Answer 1

To solve the system of equations using the substitution method, let's first look at the two equations:

1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)

We can start by manipulating Equation 1 to express \( y \) in terms of \( x \):

From Equation 1:
\[
3y = 4 - 2x
\]
\[
y = \frac{4 - 2x}{3}
\]

Now we substitute this expression for \( y \) in Equation 2. However, we will first simplify Equation 2.

Notice that we can simplify Equation 2. Divide the entire equation by -2:
\[
2x + 3y = 4
\]

As we can see, Equation 2 simplifies to the same form as Equation 1. Therefore, both equations are equivalent.

This means that the system of equations has infinitely many solutions, as they represent the same line in the coordinate plane.

Thus, the solution for the system is:

**infinite solutions**