Question

what is the equation of the line in point - slope form? use the red point in your equation. write your answer using integers, proper fractions, and improper fractions in simplest form. y - \square = \square (x - \square)

Explanation:

Step1: Identify the red point's coordinates

Looking at the graph, the red point is at \((-100, -600)\)? Wait, no, let's check the grid. Wait, the x-axis: each grid is 100? Wait, the red point: let's see, the x-coordinate: from 0, moving left 1 grid (since each grid is 100? Wait, no, the labels are -1000, -800, ..., 0, 200, etc. Wait, maybe each grid is 100? Wait, the red point: let's check the y-coordinate. The red point is at y = -600? Wait, no, the vertical lines: the y-axis has 0, 200, 400, etc. Wait, maybe the red point is at \((-100, -600)\)? Wait, no, let's find two points on the line. The line passes through (0, -300)? Wait, no, when x=0, y is -300? Wait, no, let's take two points. Let's see, when x=0, y is -300? Wait, no, the line crosses the origin? Wait, no, the line passes through (0, -300) and (100, 0)? Wait, no, let's calculate the slope. Wait, the red point: looking at the graph, the red point is at \((-100, -600)\)? Wait, no, maybe the red point is at \((-100, -600)\)? Wait, no, let's check the coordinates. Wait, the x-axis: each small square is 100? So from 0, moving left 1 square (x=-100), and y: from -600, the red point is at y=-600? Wait, no, the red point is at (-100, -600)? Wait, no, let's find two points. Let's take (0, -300) and (100, 0). Wait, slope \(m = \frac{0 - (-300)}{100 - 0} = \frac{300}{100} = 3\). Wait, that makes sense. So the slope is 3. Now, the red point: let's see, if the red point is (-100, -600), then using point-slope form \(y - y_1 = m(x - x_1)\), where \(m = 3\), \(x_1 = -100\), \(y_1 = -600\). So \(y - (-600) = 3(x - (-100))\), which simplifies to \(y + 600 = 3(x + 100)\). Wait, but let's confirm the slope. Let's take another point. If x=0, y should be -300 (since slope 3, from x=-100, y=-600: when x increases by 100, y increases by 300, so -600 + 300 = -300, which matches x=0, y=-300. Then x=100, y=0. So slope is (0 - (-300))/(100 - 0) = 300/100 = 3. So slope m=3. The red point is (-100, -600). So point-slope form is \(y - (-600) = 3(x - (-100))\), which is \(y + 600 = 3(x + 100)\). Wait, but the problem's form is \(y - \square = \square(x - \square)\). So \(y - (-600) = 3(x - (-100))\) becomes \(y + 600 = 3(x + 100)\), so in the form \(y - (-600) = 3(x - (-100))\), so the blanks are -600, 3, -100. Wait, let's check: \(y - (-600) = 3(x - (-100))\) is \(y + 600 = 3(x + 100)\), which is correct.

Step2: Write the point-slope form

Point-slope form is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the red point, and \(m\) is the slope. We found \(m = 3\), \(x_1 = -100\), \(y_1 = -600\). So substituting, we get \(y - (-600) = 3(x - (-100))\), which simplifies to \(y + 600 = 3(x + 100)\). So in the given form \(y - \square = \square(x - \square)\), the first blank is -600, the second is 3, the third is -100.

Answer:

\(y - (-600) = 3(x - (-100))\) so the blanks are \(-600\), \(3\), \(-100\)

Wait, but let's recheck the red point. Maybe the red point is at (-100, -600). Let's confirm the slope. If we take (0, -300) and (-100, -600), the slope is \(\frac{-300 - (-600)}{0 - (-100)} = \frac{300}{100} = 3\), which is correct. So yes, the point-slope form using the red point \((-100, -600)\) and slope 3 is \(y - (-600) = 3(x - (-100))\), so \(y + 600 = 3(x + 100)\). So the answer is \(y - (-600) = 3(x - (-100))\), so the blanks are -600, 3, -100.