Question

Use the substitution method to solve the system 2x+3y=4
2
x
+
3
y
=
4
, −4x−6y=−8
−
4
x
−
6
y
=
−
8
.(1 point)
Responses

no solution
no solution

infinite solutions
infinite solutions

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

(−1,2)

Answer 1

To use the substitution method to solve the given system of equations, we can start with the first equation:

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

From this equation, we can solve for one variable in terms of the other. Let's solve for \(y\):

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

Now we substitute \(y\) into the second equation:

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

Substituting \(y\) gives us:

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

Now we simplify this equation:

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

Calculate \(6(4 - 2x)\):

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

Putting it back into the equation:

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

Now let's break down the fraction:

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

Combine like terms:

\[
-8 = -8
\]

This is a true statement indicating that both equations are actually dependent (the second equation is just a multiple of the first). This means that we have infinitely many solutions.

Thus, the system has **infinite solutions**.

In conclusion, the correct response is:

**infinite solutions**.