Question

look at this graph: graph of a line with a red point at (0, 30) or similar? wait, the red point is at (0, 30? wait, the y-axis has 20, 40, so the red point is at (0, 30? wait, the grid: x from -100 to 40, y from -100 to 100. the red point is at (0, 30? wait, the line is decreasing. then the 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.

Explanation:

Step1: Identify the red point

The red point is at \((0, 30)\) (since it's on the y - axis, \(x = 0\) and \(y=30\)).

Step2: Find another point on the line

Let's find the slope. We can take two points. Let's take the red point \((0,30)\) and another point, say when \(x = 100\), let's see the y - value. From the graph, the line is decreasing. Let's calculate the slope. Let's take two points: \((0,30)\) and \((100,20)\) (we can estimate the change). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{20 - 30}{100 - 0}=\frac{- 10}{100}=-\frac{1}{10}\). Wait, maybe a better way. Let's take two points with integer coordinates. Let's see, when \(x = 0\), \(y = 30\); when \(x = 100\), \(y=20\). So the slope \(m=\frac{20 - 30}{100 - 0}=-\frac{1}{10}\).

Step3: Use point - slope form

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 and \(m\) is the slope. We use the red point \((0,30)\), so \(x_1 = 0\), \(y_1=30\) and \(m =-\frac{1}{10}\). Plugging into the formula: \(y - 30=-\frac{1}{10}(x - 0)\), which simplifies to \(y - 30=-\frac{1}{10}x\). Wait, maybe I made a mistake in the slope. Let's re - calculate. Let's take two points: \((0,30)\) and \((100,20)\). The change in \(y\) is \(20 - 30=- 10\), change in \(x\) is \(100 - 0 = 100\), so slope \(m=\frac{-10}{100}=-\frac{1}{10}\). Alternatively, let's take \((0,30)\) and \((- 100,40)\). Change in \(y\): \(40 - 30 = 10\), change in \(x\): \(-100-0=-100\), so \(m=\frac{10}{-100}=-\frac{1}{10}\). So the slope is \(-\frac{1}{10}\) and the point is \((0,30)\). So the point - slope form is \(y - 30=-\frac{1}{10}(x - 0)\) or \(y - 30=-\frac{1}{10}x\). Wait, maybe the red point is \((0,30)\)? Wait, looking at the graph, the red point is at \((0,30)\)? Wait, the y - axis has 0,20,40... Wait, the red point is at (0, 30)? Wait, the grid: the y - axis, the red point is at (0, 30)? Wait, the vertical axis: 0,20,40... Wait, the red point is at (0, 30)? Wait, maybe I misread. Let's check the graph again. The red point is at (0, 30)? Wait, the y - axis: 0,20,40... So between 20 and 40, it's 30. So the red point is (0, 30). Then, let's find another point. Let's take x = 100, then y = 20 (since the line is decreasing). So slope \(m=\frac{20 - 30}{100 - 0}=-\frac{1}{10}\). So the point - slope form using the point (0,30) is \(y - 30=-\frac{1}{10}(x - 0)\), which is \(y - 30=-\frac{1}{10}x\).

Answer:

\(y - 30=-\frac{1}{10}x\) (or \(y - 30=-\frac{1}{10}(x - 0)\))