Let's go through each section of the worksheet and solve the problems step by step, filling in the blanks and explaining the solutions.
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1. When You Have a Graph
Example: A line is drawn on a coordinate grid.
What to Do...
> Count
rise over
run on the graph.
✔ Explanation:
To find the slope from a graph, you use the "rise over run" method:
-
Rise = change in y (vertical change)
-
Run = change in x (horizontal change)
You pick two points on the line, count how many units you move up or down (rise), and how many units left or right (run). Then divide rise by run.
For this example:
- The line goes from point (1, 3) to (2, -1), for instance.
- Rise: from y = 3 to y = -1 → ↓4 units → rise = -4
- Run: from x = 1 to x = 2 → →1 unit → run = 1
- Slope = $ \frac{-4}{1} = -4 $
So, the slope is
-4, which matches the steep negative slope shown.
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2. Two Points: (2, 4) and (0, -7)
Use the slope formula:
$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$
Let’s assign:
- $ (x_1, y_1) = (2, 4) $
- $ (x_2, y_2) = (0, -7) $
Now plug into the formula:
$$
m = \frac{-7 - 4}{0 - 2} = \frac{-11}{-2} = \frac{11}{2}
$$
✔ Answer: The slope is $ \boxed{\frac{11}{2}} $
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3. Equation in Standard Form: $ 2x + 3y = 10 $
What to Do...
> Change equation to
slope-intercept form.
✔ Explanation:
Slope-intercept form is $ y = mx + b $. We need to solve for $ y $:
Start with:
$$
2x + 3y = 10
$$
Subtract $ 2x $ from both sides:
$$
3y = -2x + 10
$$
Divide every term by 3:
$$
y = -\frac{2}{3}x + \frac{10}{3}
$$
✔ So, the slope $ m = -\frac{2}{3} $
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4. Equation in Slope-Intercept Form: $ y = 3x + 7 $
What to Do...
> Just find “m”!
✔ Explanation:
In slope-intercept form $ y = mx + b $, the coefficient of $ x $ is the slope $ m $.
So here, $ m = 3 $
✔ Answer: The slope is $ \boxed{3} $
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✔ Final Answers Filled In:
| WHEN YOU HAVE... | EXAMPLE | WHAT TO DO... |
|------------------|--------|----------------|
| A Graph | (Graph shown) | Count
rise over
run on the graph |
| Two Points | (2, 4), (0, -7) | Use the slope formula: $ m = \frac{y_2 - y_1}{x_2 - x_1} $ → $ m = \frac{11}{2} $ |
| The equation in standard form $ ax + by = c $ | $ 2x + 3y = 10 $ | Change equation to
slope-intercept form → $ y = -\frac{2}{3}x + \frac{10}{3} $ → $ m = -\frac{2}{3} $ |
| The equation in slope-intercept form $ y = mx + b $ | $ y = 3x + 7 $ | Just find “m”! → $ m = 3 $ |
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🔍 Summary of Slopes Found:
1. From graph:
Negative slope, approximately $-4$ (based on visual).
2. From two points: $ \boxed{\frac{11}{2}} $
3. From standard form: $ \boxed{-\frac{2}{3}} $
4. From slope-intercept form: $ \boxed{3} $
Let me know if you'd like this formatted as a completed worksheet!
Parent Tip: Review the logic above to help your child master the concept of slopes of lines worksheet.
