Question

Use the substitution method to solve the system 2x+3y=4 , −4x−6y=−8 .(1 point) Responses no solution no solution infinite solutions infinite solutions (2−1) left parenthesis 2 comma negative 1 right parenthesis (−1,2)

Answer 1

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

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

Let's manipulate the first equation to express \( y \) in terms of \( x \):

1. From the first equation, we can isolate \( y \):

\[
3y = 4 - 2x
\]

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

Now, let's substitute this expression for \( y \) back into the second equation.

2. Substitute \( y \) in the second equation:

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

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

\[
-12x - 6(4 - 2x) = -24
\]

Now distribute \( -6 \):

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

Now combine like terms:

\[
-24 = -24
\]

This is a true statement, which means the two equations represent the same line. Thus, there are infinitely many solutions to this system.

Therefore, the answer is:

**Infinite solutions.**