Question

Use the substitution method to solve the system 2x+3y=4, −4x−6y=−8.(1 point)%0D%0AResponses%0D%0A%0D%0Ano solution%0D%0Ano solution%0D%0A%0D%0A(2−1)%0D%0Aleft parenthesis 2 comma negative 1 right parenthesis%0D%0A%0D%0Ainfinite solutions%0D%0Ainfinite solutions%0D%0A%0D%0A(−1,2)

Answer 1

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

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

First, let's solve Equation 1 for one variable, say \( y \):

\[
2x + 3y = 4
\]

Subtract \( 2x \) from both sides:

\[
3y = 4 - 2x
\]

Now, divide by 3:

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

Next, we'll substitute this expression for \( y \) into Equation 2:

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

Substituting \( y \) gives:

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

Multiply through by 3 to eliminate the fraction:

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

Distributing \( -6 \):

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

Now, notice that the \( -12x \) and \( 12x \) cancel out:

\[
-24 = -24
\]

This is a true statement, indicating that the two equations are actually equivalent and therefore give us the same line. Thus, the system has **infinite solutions**.

The final answer is **infinite solutions**.