Question

Four friends analyzed the system of equations 6y=(2/3) x+5 and 6y−(1/2)x=5. Remy, Viola, Burke, and Florence each reached a different conclusion about the system of equations.

Which friend analyzed the system correctly and drew the correct conclusion?
A) Viola noticed that each equation has the same slope but a different y-intercept, so she concluded that the system has no solution.
B) Remy noticed that there are two equations and two unknowns, and that each equation has a different slope, so she concluded that the system probably has at least one solution.
C) Burke noticed that each equation has the same slope and same y-intercept, so she concluded that there must be infinitely many solutions.
D) Florence noticed that the system has the same number of unknowns and equations, so she concluded that the system must have at least one solution.

Answer 1

Remy made the correct conclusion regarding the system of equations by identifying that each equation has a different slope and therefore, the system has exactly one solution.

Step-by-step explanation:

The student asked which friend analyzed the system of equations correctly. The two given equations are 6y=(2/3)x+5 and 6y−(1/2)x=5. To determine the correct conclusion, we need to analyze the slope and y-intercept of each line.

Let's rewrite the equations in the slope-intercept form y=mx+b to clearly see the slope (m) and the y-intercept (b):

• For the first equation, y=((2/3)/6)x + (5/6) which simplifies to y=(1/9)x + (5/6)
• For the second equation, y=((1/2)/6)x + (5/6) which simplifies to y=(1/12)x + (5/6)

Now we see that the slopes of both the equations are different, (1/9) ≠ (1/12), and they have the same y-intercept (5/6). Therefore, these two lines intersect at only one point and the system has exactly one solution. The correct conclusion is made by Remy: there are two equations and two unknowns, and each equation has a different slope, so she concluded that the system probably has at least one solution.