Question

Use the substitution method to solve the system 2x+3y=4 , −4x−6y=−8 .

Answer 1

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

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

First, let's solve one of the equations for one variable. We can solve Equation 1 for \( y \):

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

Now that we have \( y \) expressed in terms of \( x \), we can substitute this expression into Equation 2:

Substituting \( y = \frac{4 - 2x}{3} \) into Equation 2:

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

Simplifying the left side:

\[
-4x - \frac{24 - 12x}{3} = -8
\]
\[
-4x - 8 + 4x = -8
\]

The \( -4x \) and \( +4x \) cancel out:

\[
-8 = -8
\]

This is a true statement, which indicates that the two equations are dependent — they represent the same line. Thus, every point on the line is a solution.

To express the solution, we rewrite \( y \) in terms of \( x \):

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

So the solution set is:

\[
\{(x, \frac{4 - 2x}{3}) \mid x \in \mathbb{R}\}
\]

This indicates that there are infinitely many solutions that lie along the line represented by \( 2x + 3y = 4 \).