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 use the substitution method to solve the system of equations given by:

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

First, we can solve one of the equations for one variable in terms of the other. Let's solve the first equation for \( y \):

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

Now we can substitute this expression for \( y \) into the second equation:

\[
-4x - 6y = -8
\]

Substituting \( y \):

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

Multiplying through by 3 to eliminate the fraction:

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

Notice that \( -12x + 12x = 0 \), simplifying gives us:

\[
-24 = -24
\]

This statement is always true, meaning that the two equations are essentially the same. Thus, there are infinitely many solutions that satisfy both equations.

In conclusion, the correct response is:

**infinite solutions**.