Question

1: find the slope: (3, 19), (-1, -13)
2: find the slope: graph of a line on a coordinate plane

Explanation:

Step1: Identify points on the line

From the graph, we can pick two clear points. Let's take \((0, 1)\) and \((1, 4)\) (or other visible points, but these are easy to identify). Wait, actually, looking at the grid, let's check the coordinates. Let's take two points: say \((-1, -3)\) and \((0, 1)\)? Wait, no, let's look again. Wait, maybe better to use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's pick two points: for example, when \(x = 0\), \(y = 1\)? Wait, no, maybe the points are \((-2, -5)\), \((-1, -1)\), \((0, 3)\), \((1, 7)\)? Wait, no, maybe I misread. Wait, let's check the vertical and horizontal changes. From one point to another, say from \((-1, -1)\) to \((0, 3)\): the change in \(y\) is \(3 - (-1)=4\), change in \(x\) is \(0 - (-1)=1\), so slope is \(4/1 = 4\)? Wait, no, maybe another pair. Wait, let's take two points: let's see, the line passes through \((-1, -3)\) and \((0, 1)\)? No, maybe the correct points are \((-1, -3)\) and \((1, 5)\)? Wait, no, let's do it properly. Wait, the graph: let's take two points with integer coordinates. Let's take \((-1, -3)\) and \((0, 1)\): no, the difference in \(y\) is \(1 - (-3)=4\), difference in \(x\) is \(0 - (-1)=1\), so slope \(4\). Wait, but maybe the points are \((-2, -7)\), \((-1, -3)\), \((0, 1)\), \((1, 5)\), \((2, 9)\). So the slope between \((-1, -3)\) and \((0, 1)\) is \(\frac{1 - (-3)}{0 - (-1)}=\frac{4}{1}=4\). Wait, but maybe I made a mistake. Wait, let's check another pair: \((-1, -3)\) and \((1, 5)\): \(\frac{5 - (-3)}{1 - (-1)}=\frac{8}{2}=4\). Yes, so the slope is \(4\). Wait, but maybe the correct slope is 4? Wait, no, maybe I misread the graph. Wait, let's look again. Wait, the problem is "Find the slope" for the line in Puzzle #3, part 2. Let's use the slope formula. Let's pick two points on the line. Let's take \((-1, -3)\) and \((0, 1)\): \(y_2 - y_1 = 1 - (-3)=4\), \(x_2 - x_1 = 0 - (-1)=1\), so \(m = 4/1 = 4\). Alternatively, \((0, 1)\) and \((1, 5)\): \(5 - 1 = 4\), \(1 - 0 = 1\), slope \(4\). So the slope is 4.

Wait, but maybe the points are different. Wait, let's check the grid again. Each square is 1 unit. So from a lower point to a higher point, moving right 1, up 4. So rise over run is 4/1 = 4. So the slope is 4.

Step2: Confirm with slope formula

Using two points, say \((x_1, y_1)=(-1, -3)\) and \((x_2, y_2)=(0, 1)\). Then slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{1 - (-3)}{0 - (-1)}=\frac{4}{1}=4\).

Answer:

4