Which statements about the system are true? Select TWO options.

y = 1/3x - 4
Зу - х = -7

A. The system has one solution.

B. The system consists of parallel lines.

C. Both lines have the same slope.

D. Both lines have the same y-intercept.

E. The equations represent the same line.

2 Answers:

Parallel lines always have equal slope, but different y intercepts.

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Explanation:

Let's solve the second equation for y

3y - x = -7

3y = -7+x

3y = x-7

y = (x-7)/3

y = x/3 - 7/3

y = (1/3)x - 7/3

The equation is in y = mx+b form with m = 1/3 as the slope and b = -7/3 as the y intercept. We see that the first equation, where y was already isolated, also has a slope of m = 1/3. The two equations of this system have the same slope. Choice C is one of the answers.

However, they don't have the same y intercept. The first equation has y intercept b = -4, while the second has b = -7/3. This means that they do not represent the same line. They need to have identical slopes, and identical y intercepts (though the slope can be different from the y intercept of course) in order to have identical lines. So we can rule out choice D and E because of this.

Because the two equations have the same slope, but different y intercepts, this means the lines are parallel. Choice B is the other answer.

Parallel lines never touch or intersect, which in turn means there is no solution point. A solution point is where the lines cross. We can rule out choice A.

I recommend using your graphing calculator, Desmos, GeoGebra, or any graphing tool (on your computer or online) to graph each equation given. You should see two parallel lines forming. I used GeoGebra to make the graph shown below.