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

Question

Use the substitution method to solve the system 2x + 3y = 4, -4x - 6y = -8
A. (-1, 2)
B. no solution
C. (2, -1)
D. infinite solutions

GPT-4o mini · Answer

To solve the system of equations using the substitution method, we first need to express one variable in terms of the other using either of the two equations.

Let's start with the first equation:

$2 x + 3 y = 4$

We can solve for $y$ :

$3 y = 4 - 2 x$ $y = \frac{4 - 2 x}{3}$

Now, we will substitute this expression for $y$ into the second equation:

$- 4 x - 6 y = - 8$

Substituting $y$ :

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

Multiplying through:

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

To eliminate the fraction, we can multiply the entire equation by 3:

$- 12 x - (24 - 12 x) = - 24$

Distributing:

$- 12 x - 24 + 12 x = - 24$

Now, simplifying:

$- 24 = - 24$

This statement is always true, indicating that the two equations are equivalent. Therefore, the system has infinitely many solutions.

The answer is D. infinite solutions.