Use the substitution method to solve the system 2x+3y=4

Question

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.