Question

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

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

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

no solution
no solution

infinite solutions

Answer 1

To solve the system of equations using the substitution method, let's start with the equations provided:

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

First, we'll manipulate the first equation to express one variable in terms of the other. Let's solve for \( y \) in terms of \( x \):

From equation (1):
\[ 3y = 4 - 2x \]
\[ y = \frac{4 - 2x}{3} \]
Now we can substitute this expression for \( y \) into the second equation.

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

Now, multiply \( -6 \) by the expression:
\[ -4x - \frac{24 - 12x}{3} = -8 \]

Multiply through by 3 to eliminate the fraction:
\[ -12x - (24 - 12x) = -24 \]
\[ -12x - 24 + 12x = -24 \]

Combining the terms:
\[ -24 = -24 \]

This is a true statement that doesn't depend on either variable, which indicates that the two equations represent the same line. Therefore, the system has infinite solutions.

Thus, the answer is:

**infinite solutions**