Thursday, May 29, 2025
Equation of a Line
1. Introduction
Hook: Imagine you're designing a roller coaster. Engineers need to calculate the exact slope of each section to ensure a smooth ride. How do they determine the steepness of each track? The answer lies in the equation of a line!
Objective:
By the end of this lesson, you'll be able to:
✔ Write the equation of a line in slope-intercept form (y = mx + b).
✔ Find the slope and y-intercept from a graph or two points.
✔ Convert between different forms of linear equations.
2. Prerequisite Knowledge Check
Before we begin, make sure you're familiar with:
Plotting points on a graph (Quick Review)
Solving for a variable in an equation (Quick Review)
Understanding slope (rise over run) (Quick Review)
3. Core Concept Explanation (I Do – Teacher Models)
Key Definitions:
Slope-Intercept Form: y = mx + b
m = slope (steepness of the line)
b = y-intercept (where the line crosses the y-axis)
Example 1: Writing an Equation from a Graph
📈 Given: A line passes through (0, 2) and has a slope of 3.
✅ Solution:
b (y-intercept) = 2 (since it crosses the y-axis at 2)
m (slope) = 3
Equation: y = 3x + 2
Example 2: Finding Slope from Two Points
📍 Given: Points (1, 5) and (3, 9)
✅ Solution:
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Slope (m) = (9 - 5)/(3 - 1) = 4/2 = 2
Use one point to find b:
5 = 2(1) + b → b = 3
Equation: y = 2x + 3
Example 3: Converting to Slope-Intercept Form
📝 Given: 4x + 2y = 8
✅ Solution:
Solve for y:
2y = -4x + 8
y = -2x + 4
4. Guided Practice (We Do – Teacher & Students Together)
Let's solve this together:
Problem: Find the equation of a line passing through (2, 8) with a slope of -1.
🔹 Step 1: What's the general form? (y = mx + b)
🔹 Step 2: Plug in the slope (m = -1) → y = -x + b
🔹 Step 3: Use the point (2, 8) → 8 = -(2) + b
🔹 Step 4: Solve for b → b = 10
✅ Final Equation: y = -x + 10
(Try two more examples together! Ask the students to give the point and the slope.)
5. Peer Practice (We Do – Students Work Together)
With a partner, solve:
Problem: A line passes through (0, -4) with slope 1/2. What's the equation?
Hint: Start with y = mx + b and find b.
Problem: Find the equation for points (-1, 6) and (1, 2).
Hint: First, find slope (m), then b.
Problem: Rewrite 3x-y = 5 in slope-intercept form.
Hint: Isolate y!
💬 Discuss: Did you and your partner get the same answer?
6. Independent Practice (You Do – Students Try Alone)
Try these on your own (solutions at the end):
Find the equation of a line with slope 4 and y-intercept -7.
A line passes through (3, 10) and (5, 16). Find its equation.
Rewrite 5x + 2y = 10 in slope-intercept form.
7. Common Mistakes & Troubleshooting
🚨 Watch out for:
Forgetting to simplify slope (e.g., 6/3 should be 2, not left as a fraction. Reduce to the lowest term.).
Mixing up x and y when plugging in points.
Misidentifying the y-intercept (it's where x = 0).
8. Real-World Application
Engineers use linear equations to design roads, bridges, and video game graphics! Knowing the slope helps them calculate angles for safety and efficiency.
Economics & Business: Profit Forecasting. Businesses predict profit, set sales targets, and determine break-even points.
Physics: Motion at Constant Speed. Calculates distance traveled over time (e.g., cars, robots, athletes).
Engineering: Hooke’s Law (Springs). Designs springs for machines, scales, or shock absorbers. Ensures springs function safely in products like car suspensions.
Medicine: Drug Dosage Effects. Determines how dosage impacts a patient’s response (e.g., pain relief).
Agriculture: Crop Yield vs. Fertilizer. Optimizes fertilizer use to maximize harvests.
9. Summary & Key Takeaways
✨ Remember:
Slope-intercept form: y = mx + b
Find the slope (m) using two points or from a graph.
Y-intercept (b) is where the line crosses the y-axis.
Always double-check your work by plugging in a point!
10. Practice & Extension
Extra Practice:
Find the equation for a line with slope -2 passing through (4, -3).
Convert 6x - 3y = 12 to slope-intercept form.
Challenge Question:
A line passes through (0, 5) and is parallel to y = 2x - 1. What's its equation?
💬 Post your answers below and discuss!
Problems on the Slope of a Line
Find the slope of the line passing through (2, 5) and (4, 9).
Solution:
Click to answer
Determine the slope of the line passing through (-1, 3) and (5, -7).
Solution:
Click to answer
A line has a slope of and passes through (6, -2). Find another point on the line.
Solution:
Using , let :
→
Find the slope of the line .
Solution:
Rewrite in slope-intercept form:
→ → Slope Click to answer
A line passes through (0, 4) and has a slope of -5. Write its equation.
Solution:
Using , :
Equation: Click to answer
Two points on a line are (3, -1) and (7, k). If the slope is 2, find k.
Solution:
→ → Click to answer
Find the slope of a line perpendicular to .
Solution:
Perpendicular slope = negative reciprocal:
Click to answer
If a line is horizontal and passes through (4, -2), what is its slope?
Solution:
Horizontal lines have slope Click to answer
A line has an undefined slope and passes through (5, -3). Write its equation.
Solution:
Lines with undefined slope are vertical:
Equation: Click to answer
The slope between (a, 8) and (-3, 2) is -2. Find the value of a.
Solution:
→ → Click to answer
Find the slope of the line passing through (2, 5) and (4, 9).
Solution:
Click to answer
Determine the slope of the line passing through (-1, 3) and (5, -7).
Solution:
Click to answer
A line has a slope of and passes through (6, -2). Find another point on the line.
Solution:
Using , let :
→
Find the slope of the line .
Solution:
Rewrite in slope-intercept form:
→ → Slope Click to answer
A line passes through (0, 4) and has a slope of -5. Write its equation.
Solution:
Using , :
Equation: Click to answer
Two points on a line are (3, -1) and (7, k). If the slope is 2, find k.
Solution:
→ → Click to answer
Find the slope of a line perpendicular to .
Solution:
Perpendicular slope = negative reciprocal:
Click to answer
If a line is horizontal and passes through (4, -2), what is its slope?
Solution:
Horizontal lines have slope Click to answer
A line has an undefined slope and passes through (5, -3). Write its equation.
Solution:
Lines with undefined slope are vertical:
Equation: Click to answer
The slope between (a, 8) and (-3, 2) is -2. Find the value of a.
Solution:
→ → Click to answer
_______________________________________________________________________
Practice Problems with Partial Solutions
Find the slope of the line passing through (2, 5) and (4, 9).
Solution:Determine the slope of the line passing through (-1, 3) and (5, -7).
Solution:A line has a slope of and passes through (6, -2). Find another point on the line.
Solution:
Using , let :
→ → →
Another point:Find the slope of the line .
Solution:
Rewrite in slope-intercept form:
→
SlopeA line passes through (0, 4) and has a slope of -5. Write its equation.
Solution:
Using , :
Equation:Two points on a line are (3, -1) and (7, k). If the slope is 2, find k.
Solution:
→ → → .Find the slope of a line perpendicular to .
Solution:
Perpendicular slope = negative reciprocal:If a line is horizontal and passes through (4, -2), what is its slope?
Solution:
Horizontal lines have slopeA line has an undefined slope and passes through (5, -3). Write its equation.
Solution:
Lines with undefined slope are vertical:
Equation:The slope between (a, 8) and (-3, 2) is -2. Find the value of a.
Solution:
→
→
11. Further Resources
📚 Learn More:
Next Lesson: Point-Slope Form & Standard Form 🚀