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 first have the two equations:

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

Notice that the second equation can be simplified. Divide every term in the second equation by -2:

\[
-4x - 6y = -8 \implies 2x + 3y = 4
\]

Now our system looks like this:

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

Both equations are actually the same line. This means the system has infinitely many solutions along the line represented by the equation \( 2x + 3y = 4 \).

To express the solutions parametrically, we can solve for \( y \) in terms of \( x \):

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

Thus, we have:

\[
\text{Solutions: } (x, y) = \left( x, \frac{4 - 2x}{3} \right) \quad \text{for any real number } x.
\]

In conclusion, the solution set to the system of equations is infinite, represented as:

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