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 two points on the line

From the graph, we can see that the line passes through the point \((400, 0)\) and the red point \((800, -1000)\) (assuming the red point is at \(x = 800\), \(y=- 1000\), and the x - intercept is at \((400,0)\)). Let's also check the y - intercept? Wait, no, let's use two clear points. Wait, actually, the line also passes through \((0, 1000)\)? Wait, no, looking at the axes: the y - axis is vertical (labeled \(y\)) and x - axis is horizontal (labeled \(x\)). Wait, the top arrow is for \(y\) (with - 1000, - 800,...) and the bottom arrow is for \(x\) (with 0, 200,...). Wait, maybe I mixed up the axes. Let's re - identify: the vertical axis is \(y\) (with positive upwards? Wait, no, the labels: on the vertical axis, the top is - 1000, - 800,... and the bottom is 0, 200,...? Wait, that's unusual. Wait, maybe the vertical axis is \(y\) with positive upwards, but the numbers are negative at the top? No, that doesn't make sense. Wait, maybe the graph has the \(y\) - axis with positive direction upwards, but the line is going from \((0,1000)\) (top) to \((800, - 1000)\) (bottom right). Wait, let's find two points:

Point 1: When \(x = 0\), \(y=1000\) (from the top of the line, when \(x = 0\), \(y\) is 1000? Wait, no, the line starts at the top left (maybe \(x = 0\), \(y = 1000\)) and goes to the red point \((800,-1000)\). Also, it crosses the \(x\) - axis at \((400,0)\). Let's confirm: the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((x_1,y_1)=(0,1000)\) and \((x_2,y_2)=(400,0)\). Then \(m=\frac{0 - 1000}{400 - 0}=\frac{- 1000}{400}=-\frac{5}{2}\). Let's check with \((400,0)\) and \((800,-1000)\): \(m=\frac{-1000 - 0}{800 - 400}=\frac{-1000}{400}=-\frac{5}{2}\). So the slope \(m =-\frac{5}{2}\).

The red point is \((800,-1000)\). The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line. We can use the red point \((x_1,y_1)=(800,-1000)\) and \(m =-\frac{5}{2}\).

Step2: Write the point - slope form

The point - slope formula is \(y - y_1=m(x - x_1)\). Substituting \(y_1=-1000\), \(m =-\frac{5}{2}\), and \(x_1 = 800\) into the formula:

\(y-(-1000)=-\frac{5}{2}(x - 800)\)

Simplify the left - hand side: \(y + 1000=-\frac{5}{2}(x - 800)\)

Wait, but the problem asks for the form \(y-\square=\square(x - \square)\). So we need to rewrite \(y + 1000=-\frac{5}{2}(x - 800)\) as \(y-(-1000)=-\frac{5}{2}(x - 800)\)

So the values are: \(y-(-1000)=-\frac{5}{2}(x - 800)\)

Let's verify the slope again. From \((0,1000)\) to \((800,-1000)\): change in \(y=-1000 - 1000=-2000\), change in \(x = 800-0 = 800\), slope \(m=\frac{-2000}{800}=-\frac{5}{2}\), which matches.

Answer:

\(y - (-1000)=-\frac{5}{2}(x - 800)\) or \(y+1000 =-\frac{5}{2}(x - 800)\) (in the form \(y-\square=\square(x - \square)\), it is \(y-(-1000)=-\frac{5}{2}(x - 800)\))