Question

3)
graph of a line on a coordinate plane
5)
graph of a line on a coordinate plane
7)
graph of a line on a coordinate plane

Explanation:

Problem 3:

Step1: Identify two points on the line

We can see that the line passes through \((0, -1)\) (the y-intercept) and \((-0.5, 0)\) (x-intercept), or another point like \((0, -1)\) and \((1, -5)\) (by moving 1 unit right and 4 units down? Wait, let's check the slope. From \((0, -1)\) to \((-1, 3)\)? Wait, maybe better to count the rise over run. Let's take two clear points. Let's see, when \(x = 0\), \(y=-1\). When \(x = -1\), \(y = 3\)? Wait, no, the line is steep. Wait, maybe the slope is -5? Wait, no, let's check the grid. Each square is 1 unit. Let's take two points: \((0, -1)\) and \((-1, 4)\)? No, maybe I made a mistake. Wait, the line passes through (0, -1) and (-0.2, 0)? No, maybe the slope is -5. Wait, let's calculate the slope between (0, -1) and (-1, 4)? No, that doesn't seem right. Wait, maybe the line is \(y = -5x - 1\)? Wait, no, when x=0, y=-1. When x=-1, y=4? No, that's not matching. Wait, maybe the line is \(y = -5x - 1\)? Wait, let's check the graph again. The line is going from the top left to bottom right, passing through (0, -1) and (-1, 4)? No, maybe I misread. Wait, maybe the slope is -5. Let's see, from (0, -1) to (1, -6)? No, that's not. Wait, maybe the correct equation is \(y = -5x - 1\)? Wait, no, let's do it properly.

Wait, let's take two points: (0, -1) and (-1, 4). Wait, the difference in y is 4 - (-1) = 5, difference in x is -1 - 0 = -1. So slope \(m = \frac{5}{-1} = -5\). So the equation is \(y = -5x - 1\)? Wait, but when x=0, y=-1, which matches. Let's check another point. If x=1, y=-5(1) -1 = -6. Does the line pass through (1, -6)? Looking at the graph, yes, it seems to. So the equation is \(y = -5x - 1\).

Step2: Write the equation

Using the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We found \(m = -5\) and \(b = -1\), so the equation is \(y = -5x - 1\).

Step1: Identify two points on the line

The line passes through (0, -1) and (2, 3). Let's calculate the slope \(m = \frac{3 - (-1)}{2 - 0} = \frac{4}{2} = 2\). The y-intercept \(b\) is -1 (since it crosses the y-axis at (0, -1)).

Step2: Write the equation

Using the slope-intercept form \(y = mx + b\), with \(m = 2\) and \(b = -1\), the equation is \(y = 2x - 1\). Wait, no, when x=1, y=1? Wait, no, let's check again. Wait, the line passes through (0, -1) and (1, 0)? Wait, no, the graph shows the line passing through (0, -1) and (2, 3). Wait, (2, 3) is 2 units right and 4 units up from (0, -1)? No, 3 - (-1) = 4, 2 - 0 = 2, so slope is 4/2 = 2. So equation is \(y = 2x - 1\). Wait, but when x=1, y=2(1) -1 = 1. Does the line pass through (1, 1)? Looking at the graph, yes, it seems to. So the equation is \(y = 2x - 1\)? Wait, no, maybe (0, -1) and (1, 0)? Wait, (1, 0) is on the line. So slope is (0 - (-1))/(1 - 0) = 1/1 = 1. Wait, I made a mistake. Let's take (0, -1) and (1, 0). Then slope is (0 - (-1))/(1 - 0) = 1. So equation is \(y = x - 1\)? Wait, when x=2, y=1? But the graph shows at x=2, y=3? No, I misread the graph. Wait, the line in problem 5: let's look again. The grid: each square is 1 unit. The line passes through (0, -1) and (2, 3). So from (0, -1) to (2, 3): change in y is 4, change in x is 2, so slope 2. So equation \(y = 2x - 1\). Wait, but when x=1, y=1. Let's check the graph: at x=1, y=1? Yes, the line passes through (1, 1). So that's correct. So the equation is \(y = 2x - 1\).

Step2: Write the equation

Slope \(m = 2\), y-intercept \(b = -1\), so \(y = 2x - 1\).

Step1: Identify two points on the line

The line passes through (0, -2) and (-4, 1). Wait, no, let's take (0, -2) and (-5, 1)? No, better to take (0, -2) and (-4, 0). Wait, the line crosses the x-axis at (-4, 0) and y-axis at (0, -2). So slope \(m = \frac{-2 - 0}{0 - (-4)} = \frac{-2}{4} = -\frac{1}{2}\). Wait, let's check: from (0, -2) to (-4, 0): change in y is 2, change in x is -4, so slope is 2/(-4) = -1/2. So the equation is \(y = -\frac{1}{2}x - 2\). Let's verify: when x=0, y=-2 (correct). When x=-4, y= -1/2*(-4) -2 = 2 -2 = 0 (correct). So that's the equation.

Step2: Write the equation

Slope \(m = -\frac{1}{2}\), y-intercept \(b = -2\), so \(y = -\frac{1}{2}x - 2\).

Answer:

\(y = -5x - 1\)

Problem 5: