A coordinate grid has a line labeled 3y = 2x - 9 passing through points (0, -3) and (3, -1). Muriel says she has written a system of two linear equations that has an infinite number of solutions. One of the equations is 3y = 2x - 9. Which could be the other equation?
A) 2y = x - 4.5
B) y = (2/3)x - 3
C) 6y = 6x - 27
D) y = (3/2)x - 4.5
Answer (1)
To determine which equation could be the other one in a system of equations with an infinite number of solutions, we need to identify whether two lines are actually the same line. This means they should have the same slope and the same y-intercept, or they can be multiples of each other.
Given the original equation:
3 y = 2 x − 9
We can convert it to the slope-intercept form ( y = m x + b ):
Isolate y :
y = 3 2 x − 3
This shows that the slope m is 3 2 and the y-intercept is − 3 .
Let's analyze the given options:
A) 2 y = x − 4.5
Solve for y :
y = 2 1 x − 2.25
This has a different slope and y-intercept, so it's not the correct choice.
B) y = 3 2 x − 3
This equation is already in slope-intercept form and matches the original: same slope 3 2 and y-intercept − 3 .
Therefore, this is the correct matching line.
C) 6 y = 6 x − 27
Divide through by 6:
y = x − 4.5
This has a different slope and y-intercept, so it’s not correct.
D) y = 2 3 x − 4.5
This has a different slope from the original equation, so it does not match.
Thus, the correct option is B) y = 3 2 x − 3 , as it represents the same line as the original equation.
