all and sid graphed this system:
$y = -\frac{1}{4}x - 1$
$y = \frac{1}{4}x - 3$
they marked its solutions with points p and r.
which statement describes ali and sid’s solutions?
\bigcirc their solutions are correct.
\bigcirc they marked the y-intercepts instead of the intersection point of the two lines.
\bigcirc they marked the y-intercepts instead of the x-intercepts.
\bigcirc they marked only the x-intercepts instead of the x - and y-intercepts.

Question

Accepted Answer

# Brief Explanations:
To determine the error in Ali and Sid's solutions, we analyze the system of equations $ y = -\frac{1}{4}x - 1 $ and $ y=\frac{1}{4}x - 3 $. The solution to a system of linear equations is the intersection point of the two lines. Let's find the intercepts and the intersection:

1. **Find $ y $-intercepts**:
   - For $ y = -\frac{1}{4}x - 1 $, when $ x = 0 $, $ y=-1 $.
   - For $ y=\frac{1}{4}x - 3 $, when $ x = 0 $, $ y = - 3 $. Wait, but looking at the graph, point $ P $ is at $ (0, - 1) $? Wait, no, maybe I misread. Wait, the options suggest they marked $ y $-intercepts instead of the intersection. Let's find the intersection:
   Set $ -\frac{1}{4}x - 1=\frac{1}{4}x - 3 $
   $ -\frac{1}{4}x-\frac{1}{4}x=-3 + 1 $
   $ -\frac{2}{4}x=-2 $
   $ -\frac{1}{2}x=-2 $
   $ x = 4 $
   Then $ y=-\frac{1}{4}(4)-1=-1 - 1=-2 $? Wait, maybe my calculation is wrong. Wait, let's re - solve:
   $ -\frac{1}{4}x - 1=\frac{1}{4}x - 3 $
   Add $ \frac{1}{4}x $ to both sides: $ - 1=\frac{2}{4}x-3 $
   Add 3 to both sides: $ 2=\frac{1}{2}x $
   Multiply both sides by 2: $ x = 4 $
   Then $ y=\frac{1}{4}(4)-3=1 - 3=-2 $. So the intersection point is $ (4,-2) $.

Now, point $ P $: Let's check the coordinates of $ P $. From the graph, $ P $ seems to be on the $ x $-axis? Wait, no, the options are about intercepts. Let's check the $ y $-intercepts:
   - For $ y = -\frac{1}{4}x - 1 $, $ y $-intercept is $ (0,-1) $
   - For $ y=\frac{1}{4}x - 3 $, $ y $-intercept is $ (0,-3) $
   But the points marked are $ P $ and $ R $. If $ P $ is $ (0, - 1) $ (the $ y $-intercept of the first line) and $ R $ is $ (-4,0) $ (the $ x $-intercept of the first line: $ 0=-\frac{1}{4}x - 1\Rightarrow x=-4 $)? Wait, the option "They marked the $ y $-intercepts instead of the intersection" or "They marked $ y $-intercepts instead of $ x $-intercepts"? Wait, the options are:
   - Their solutions are correct. No, because the solution is the intersection, not the intercepts.
   - They marked the intersection point of the two lines. No, because the intercepts are not the intersection.
   - They marked the $ y $-intercepts instead of the $ x $-intercepts. No, let's re - evaluate.
   Wait, the correct approach: The solution to the system is the intersection of the two lines. If they marked points $ P $ and $ R $, which are likely the $ y $-intercept (for one line) and $ x $-intercept (for the other), but the key is that the solution of the system is the intersection, not the intercepts. So the statement "They marked the $ y $-intercepts instead of the intersection" is correct? Wait, the options given:
   Option 1: Their solutions are correct. No.
   Option 2: They marked the $ y $-intercepts instead of the intersection. Yes, because the solution is the intersection, not the $ y $-intercepts.
   Option 3: They marked the $ y $-intercepts instead of the $ x $-intercepts. No.
   Option 4: They marked only the $ x $-intercepts instead of the $ y $-intercepts. No.

So the correct statement is "They marked the $ y $-intercepts instead of the intersection point of the two lines." But looking at the options, the closest is "They marked the $ y $-intercepts instead of the intersection" (the second option in the list: "They marked the $ y $-intercepts instead of the intersection, point of the two lines." Wait, the original options:

The options are:

- Their solutions are correct.

- They marked the $ y $-intercepts instead of the intersection, point of the two lines.

- They marked the $ y $-intercepts instead of the $ x $-intercepts.

- They marked only the $ x $-intercepts instead of the $ y $-intercepts.

So the correct answer is the second option: "They marked the $ y $-intercepts instead of the intersection, point of the two lines."

# Answer:
They marked the $ y $-intercepts instead of the intersection, point of the two lines. (The second option in the given choices)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Answer: