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 have the equations:

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

First, we can solve Equation 1 for one variable in terms of the other. Let's solve for \( y \):

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

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

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

Multiplying through by 3 to eliminate the fraction:

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

Notice that the terms involving \( x \) cancel out:

\[
-24 = -24
\]

This statement is always true, which means that the two equations are dependent. The second equation is a multiple of the first (specifically, it is equivalent to multiplying the first equation by -2):

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

Since the equations represent the same line, the system has infinitely many solutions.

Thus, the answer is:

**infinite solutions**