Question

which points lie on the line that passes through point p and is parallel to the given line? select three options. □ (-4, 2) □ (-1, 3) □ (-2, 2) □ (4, 2) □ (-5, -1)

Explanation:

Step1: Find slope of given line

Two points on given line: \((-3, -4)\) and \((4, 2)\). Slope \(m = \frac{2 - (-4)}{4 - (-3)} = \frac{6}{7}\)? Wait, no, wait. Wait, looking at the grid, the given line passes through \((0, -2)\) and \((4, 2)\)? Wait, no, let's recheck. Wait, point P is \((0, 4)\). Wait, the given line: let's take two points. From the graph, the blue line passes through \((-3, -4)\) and \((4, 2)\)? Wait, no, maybe \((0, -2)\) and \((4, 2)\): change in y is \(2 - (-2) = 4\), change in x is \(4 - 0 = 4\), so slope \(m = 1\). Wait, yes, because from \((0, -2)\) to \((4, 2)\), rise 4, run 4, slope 1. Or from \((-3, -4)\) to \((0, -2)\): rise 2, run 3? No, wait, maybe I made a mistake. Wait, point P is \((0, 4)\). The line parallel to the given line will have the same slope. Let's find the slope of the given line. Let's take two points on the given line: say \((-3, -4)\) and \((4, 2)\). Wait, no, looking at the grid, when x=0, y=-2 (the line crosses y-axis at -2). Then at x=4, y=2. So slope \(m = \frac{2 - (-2)}{4 - 0} = \frac{4}{4} = 1\). So the slope of the given line is 1. Therefore, the line through P(0,4) with slope 1 has equation \(y - 4 = 1(x - 0)\), so \(y = x + 4\).

Step2: Check each point

- For \((-4, 2)\): Plug x=-4, y should be \(-4 + 4 = 0\)? Wait, no, wait \(y = x + 4\). So x=-4: \(y = -4 + 4 = 0\). But the point is (-4,2), which is not 0. Wait, maybe I messed up the slope. Wait, let's re-examine the given line. Let's take two points: (-3, -4) and (1, 0). Wait, from (-3, -4) to (1, 0): rise 4, run 4, slope 1. Then from (1, 0) to (4, 3)? No, the point on the line is (4,2)? Wait, the graph shows a point at (4,2) on the blue line. Wait, maybe my initial slope calculation is wrong. Let's take (0, -2) and (4, 2): y increases by 4, x increases by 4, so slope 1. Then the line through P(0,4) is \(y = x + 4\). Now check each option:

1. \((-4, 2)\): \(y = -4 + 4 = 0\)? No, 2≠0. Wait, maybe I made a mistake. Wait, maybe the slope is \(\frac{3}{3}\)? Wait, let's take another pair. The given line passes through (-3, -4) and (0, -2): rise 2, run 3? No, -2 - (-4) = 2, 0 - (-3) = 3, so slope \(\frac{2}{3}\)? Wait, this is confusing. Wait, let's look at point P(0,4). The line parallel to the given line: let's find two points on the given line. Let's see, the blue line goes through (-3, -4) and (4, 2). Let's calculate the slope: \(m = \frac{2 - (-4)}{4 - (-3)} = \frac{6}{7}\)? No, that can't be. Wait, maybe the grid is 1 unit per square. So from (-3, -4) to (4, 2): horizontal distance 7, vertical distance 6. No, that's not right. Wait, maybe the given line passes through (0, -2) and (4, 2). So x from 0 to 4 (change +4), y from -2 to 2 (change +4), so slope 1. Then line through P(0,4) is y = x + 4. Now check each point:

- (-4, 2): y = -4 + 4 = 0 ≠ 2. No.
- (-1, 3): y = -1 + 4 = 3. Yes!
- (-2, 2): y = -2 + 4 = 2. Yes!
- (4, 2): y = 4 + 4 = 8 ≠ 2. No.
- (-5, -1): y = -5 + 4 = -1. Yes!

Wait, so the points that lie on the line are (-1, 3), (-2, 2), (-5, -1). Wait, but let's recheck the slope. Maybe I made a mistake in the line equation. Let's recalculate the slope of the given line. Let's take two points on the given line: (0, -2) and (4, 2). Slope \(m = \frac{2 - (-2)}{4 - 0} = \frac{4}{4} = 1\). Correct. Then the line through P(0,4) is \(y = x + 4\). Now check each point:

- (-4, 2): \(y = -4 + 4 = 0\) → 2 ≠ 0 → no.
- (-1, 3): \(y = -1 + 4 = 3\) → yes.
- (-2, 2): \(y = -2 + 4 = 2\) → yes.
- (4, 2): \(y = 4 + 4 = 8\) → 2 ≠ 8 → no.
- (-5, -1): \(y = -5 + 4 = -1\) → yes.

So the three points are (-1, 3), (-2, 2), (-5, -1). Wait, but let's check the original graph again. Wait, point P is (0,4). The line parallel to the given line: let's see the direction. The given line has a slope of 1 (since it goes up 1, right 1). So from P(0,4), going left 1, down 1: (-1, 3) – which matches. Left 2, down 2: (-2, 2) – matches. Left 5, down 5: (-5, -1) – matches. Yes, that makes sense. So the correct options are (-1, 3), (-2, 2), (-5, -1).

Answer:

(-1, 3), (-2, 2), (-5, -1)