Question

1. what is the equation of the line below?

Explanation:

Step1: Identify two points on the line

From the graph, let's find two clear points. Let's take the left - most red point: let's assume the grid has each square with side length 1. Let's say the left point is \((-5, 5)\) and the right point is \((5, 1)\) (we can also use the y - intercept and another point. Wait, actually, let's check the y - intercept. The line crosses the y - axis at \((0, 3)\) maybe? Wait, let's do it properly. Let's take two points: let's say the left red point is \((-4, 5)\) and the right red point is \((6, 1)\)? Wait, no, maybe better to use the slope formula. Let's find two points with integer coordinates. Let's see, the line passes through \((-5, 5)\) and \((5, 1)\)? Wait, no, let's count the rise and run. Let's take two points: for example, \((-4, 5)\) and \((6, 1)\) – no, maybe a better way. Let's take the point where x=-5, y = 5 and x = 5, y = 1. Wait, the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((-5, 5)\) and \((5, 1)\). Then \(m=\frac{1 - 5}{5-(-5)}=\frac{-4}{10}=-\frac{2}{5}\)? Wait, no, maybe I made a mistake. Wait, let's look at the graph again. Let's take the left red point: let's say its coordinates are \((-4, 5)\) and the right red point is \((6, 1)\). Then the change in y is \(1 - 5=-4\), change in x is \(6-(-4) = 10\), so slope \(m=\frac{-4}{10}=-\frac{2}{5}\)? Wait, no, maybe another pair. Wait, the line crosses the y - axis at \((0, 3)\). Let's take another point: when x = 5, y = 1. So the two points are \((0, 3)\) (y - intercept) and \((5, 1)\). Now, calculate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1 - 3}{5 - 0}=\frac{-2}{5}=-\frac{2}{5}\)? Wait, no, that doesn't seem right. Wait, maybe the left point is \((-5, 5)\) and the y - intercept is \((0, 3)\). Then slope \(m=\frac{3 - 5}{0-(-5)}=\frac{-2}{5}=-\frac{2}{5}\). Then the equation of the line in slope - intercept form \(y=mx + b\), where \(b\) is the y - intercept. We found \(b = 3\) (since the line crosses the y - axis at \((0, 3)\)) and \(m=-\frac{2}{5}\)? Wait, no, maybe I messed up the points. Wait, let's do it again. Let's take two points: let's say the left red point is \((-4, 5)\) and the point where x = 1, y = 3? No, wait, maybe the correct two points are \((-5, 5)\) and \((5, 1)\) is wrong. Wait, let's count the grid. Let's assume each square is 1 unit. Let's take the left red point: x=-5, y = 5 (since from the y - axis (x = 0), moving 5 units left (x=-5) and 5 units up (y = 5)). Then the right red point: x = 5, y = 1 (5 units right from x = 0, 1 unit up from y = 0). Now, slope \(m=\frac{1 - 5}{5-(-5)}=\frac{-4}{10}=-\frac{2}{5}\). The y - intercept \(b\): when x = 0, what's y? Let's plug x = 0 into the slope - intercept formula \(y=mx + b\). We can use the point \((-5, 5)\): \(5=-\frac{2}{5}(-5)+b\), \(5 = 2 + b\), so \(b = 3\). So the equation is \(y=-\frac{2}{5}x+3\)? Wait, but maybe I made a mistake in the points. Wait, another way: let's take the two points as \((-5, 5)\) and \((0, 3)\). Then slope \(m=\frac{3 - 5}{0-(-5)}=\frac{-2}{5}=-\frac{2}{5}\), and \(b = 3\) (since when x = 0, y = 3). So the equation of the line is \(y=-\frac{2}{5}x + 3\). Wait, but let's check with the other point \((5, 1)\): plug x = 5 into \(y=-\frac{2}{5}(5)+3=-2 + 3 = 1\), which matches. So that works.

Step2: Write the equation in slope - intercept form

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We found that \(m =-\frac{2}{5}\) and \(b = 3\). So the equation of the line is \(y=-\frac{2}{5}x + 3\).

Answer:

\(y =-\frac{2}{5}x+3\)