Use the substitution method to solve the system 2x+3y=4

Question

, −4x−6y=−8
.(1 point)
Responses

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

infinite solutions
infinite solutions

no solution
no solution

(−1,2)

Answer 1

To solve the system of equations using the substitution method, let's first analyze the two equations:

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

First, we can simplify Equation 2. If we multiply Equation 1 by 2, we get:

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

Now let's rewrite Equation 2:

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

Now, adding both equations together:

\[ (4x + 6y) + (-4x - 6y) = 8 - 8 \ 0 = 0 \]

This means that the two equations represent the same line, indicating that they are dependent. Therefore, there are infinitely many solutions since any point on the line described by the equation \( 2x + 3y = 4 \) is a solution to the system.

So, the answer to the system is:

Infinite solutions.