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

infinite solutions
infinite solutions

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

no solution

Answer 1

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

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

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

\[
3y = 4 - 2x
\]

Now, divide both sides by 3:

\[
y = \frac{4 - 2x}{3}
\]

Next, we'll substitute this expression for \( y \) into Equation 2 (substitution step):

\[
-4x - 6\left(\frac{4 - 2x}{3}\right) = -8
\]

To eliminate the fraction, we can multiply the entire equation by 3:

\[
3(-4x) - 6(4 - 2x) = 3(-8)
\]

Simplifying:

\[
-12x - 24 + 12x = -24
\]

The \( -12x \) and \( 12x \) terms cancel each other out:

\[
-24 = -24
\]

This statement is always true, indicating that the equations are dependent and represent the same line.

Thus, the system has **infinite solutions**.

The correct response is:
**infinite solutions**.