slope of the line
count the rise and run between any two coordinates, and find the slope of each line.
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Question

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Identify two points on the line. Let's take (0, -2) and (3, 1).
## Step2: Calculate rise (change in y) and run (change in x). Rise = $1 - (-2) = 3$, Run = $3 - 0 = 3$.
## Step3: Slope = $\frac{\text{Rise}}{\text{Run}} = \frac{3}{3} = 1$.
# Answer:
$1$

### Problem 2:
# Explanation:
## Step1: Take two points, e.g., (-5, 2) and (3, -2).
## Step2: Rise = $-2 - 2 = -4$, Run = $3 - (-5) = 8$.
## Step3: Slope = $\frac{-4}{8} = -\frac{1}{2}$.
# Answer:
$-\frac{1}{2}$

### Problem 3:
# Explanation:
## Step1: Use points (0, -3) and (3, 3).
## Step2: Rise = $3 - (-3) = 6$, Run = $3 - 0 = 3$.
## Step3: Slope = $\frac{6}{3} = 2$ (Wait, earlier thought was $\frac{4}{3}$, but rechecking: if grid is 1 unit per square, let's take (0, -3) and (3, 3): rise 6, run 3, slope 2. Maybe initial mistake. Let's correct. If points are (0, -3) and (2, 1): rise 4, run 2, slope 2? Wait, maybe better to take (0, -3) and (3, 3): 6/3=2. Or ( -1, -5) and (2, 1): rise 6, run 3, slope 2. So slope is $2$? Wait, maybe the original thought of $\frac{4}{3}$ was wrong. Let's re-express. Let's take two clear points. Suppose the line passes through (0, -3) and (3, 3). Then rise is 3 - (-3) = 6, run is 3 - 0 = 3, so slope is 6/3 = 2. So correct slope is $2$.
# Answer:
$2$

### Problem 4:
# Explanation:
## Step1: Take points (-5, 1) and (3, -5).
## Step2: Rise = $-5 - 1 = -6$, Run = $3 - (-5) = 8$? Wait, no, better points: (-5, 1) and (3, -5): no, maybe (-4, 0) and (3, -5)? Wait, let's take (-5, 1) and (3, -5): rise -6, run 8? No, maybe ( -5, 1) and (3, -5) is wrong. Let's take (-4, 0) and (3, -5): rise -5, run 7? No, better to use ( -5, 1) and (3, -5): no, maybe the line passes through (-5, 1) and (3, -5): no, let's count grid. From (-5,1) to (3,-5): down 6, right 8? No, maybe (-4,0) to (4,-6): down 6, right 8? No, maybe the slope is $-\frac{3}{4}$? Wait, let's take two points: (-5, 1) and (3, -5). Rise: -5 -1 = -6. Run: 3 - (-5) = 8. Slope: -6/8 = -3/4. Yes, that's better. So slope is $-\frac{3}{4}$? Wait, no, let's take (-4, 0) and (4, -6): rise -6, run 8: -6/8 = -3/4. So slope is $-\frac{3}{4}$.
# Answer:
$-\frac{3}{4}$

### Problem 5:
# Explanation:
## Step1: Take points (0, -3) and (3, 3). Wait, no, problem 5: the line passes through (0, -3) and (3, 3)? No, problem 5: the line goes from (0, -5) to (3, 5)? Wait, no, looking at the graph, problem 5: the line passes through (0, -3) and (3, 3)? No, problem 5's graph: let's see, the line goes from ( -1, -5) to (2, 1): rise 6, run 3, slope 2? Wait, no, problem 5: the line is steeper. Wait, maybe points (0, -5) and (3, 5): rise 10, run 3? No, that can't be. Wait, maybe (0, -3) and (2, 1): rise 4, run 2, slope 2. Or (0, -5) and (3, 5): rise 10, run 3? No, that's not. Wait, let's check the grid: each square is 1 unit. So from (0, -3) to (3, 3): 6/3=2. But problem 5: the line is steeper. Wait, maybe (0, -5) and (2, 1): rise 6, run 2, slope 3? Wait, if the line passes through (0, -5) and (2, 1), then rise is 1 - (-5) = 6, run is 2 - 0 = 2, so slope is 6/2 = 3. Ah, that's better. So slope is $3$.
# Answer:
$3$

### Problem 6:
# Explanation:
## Step1: Take points (0, 0) and (3, 6).
## Step2: Rise = $6 - 0 = 6$, Run = $3 - 0 = 3$.
## Step3: Slope = $\frac{6}{3} = 2$? Wait, no, (0,0) and (1,2): rise 2, run 1, slope 2. Or (0,0) and (3,6): 6/3=2. So slope is $2$? Wait, no, ( -2, -4) and (3, 6): rise 10, run 5, slope 2. So slope is $2$.
# Answer:
$2$

### Problem 7:
# Explanation:
## Step1: Take points (-5, 5) and (3, -5).
## Step2: Rise = $-5 - 5 = -10$, Run = $3 - (-5) = 8$? No, better points: (-5, 5) and (3, -5): rise -10, run 8? No, let's take (-5, 5) and (0, 0): rise -5, run 5, slope -1. Wait, (-5,5) to (0,0): down 5, right 5, slope -1. Then (0,0) to (3, -3): down 3, right 3, slope -1. So slope is $-1$.
# Answer:
$-1$

### Problem 8:
# Explanation:
## Step1: Take points (-5, -5) and (3, 5).
## Step2: Rise = $5 - (-5) = 10$, Run = $3 - (-5) = 8$? No, better points: (-5, -5) and (0, 0): rise 5, run 5, slope 1. Then (0,0) to (3, 3): rise 3, run 3, slope 1. So slope is $1$.
# Answer:
$1$

### Problem 9:
# Explanation:
## Step1: Take points (-5, 3) and (3, -5).
## Step2: Rise = $-5 - 3 = -8$, Run = $3 - (-5) = 8$.
## Step3: Slope = $\frac{-8}{8} = -1$.
# Answer:
$-1$

(Note: Some problems required rechecking points for accuracy. The key is to identify two clear points on the line, calculate the vertical change (rise) and horizontal change (run), then divide rise by run to get the slope.)

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QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Identify two points on the line. Let's take (0, -2) and (3, 1).

Step2: Calculate rise (change in y) and run (change in x). Rise = \(1 - (-2) = 3\), Run = \(3 - 0 = 3\).

Step3: Slope = \(\frac{\text{Rise}}{\text{Run}} = \frac{3}{3} = 1\).

Step1: Take two points, e.g., (-5, 2) and (3, -2).

Step2: Rise = \(-2 - 2 = -4\), Run = \(3 - (-5) = 8\).

Step3: Slope = \(\frac{-4}{8} = -\frac{1}{2}\).

Step1: Use points (0, -3) and (3, 3).

Step2: Rise = \(3 - (-3) = 6\), Run = \(3 - 0 = 3\).

Answer:

Problem 2: