Question

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

1. No solution
2. Infinite solutions
3. (2 - 1)
4. (-1, 2)

Answer 1

To solve the system of equations using the substitution method, we need to express one variable in terms of the other and then substitute that expression into the second equation.

The given system of equations is:

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

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

\[
3y = 4 - 2x
\]

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

Now we can substitute this expression for \(y\) in Equation 2.

Substituting \(y\) into Equation 2:

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

Now, simplify this:

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

Let's multiply out the fraction:

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

Convert it to have a common denominator:

\[
-4x - 8 + 4x = -8
\]

Now, simplifying the left side, we see:

\[
-8 = -8
\]

This statement is always true, which implies that the two equations are dependent. Hence, they describe the same line.

Therefore, the system of equations has **infinite solutions**.

The correct answer is:
**2. Infinite solutions**.