Equation of a line Mastery in 3 Steps: Watch, Try, Solve!

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.

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2. Prerequisite Knowledge Check

Before we begin, make sure you're familiar with:

1. Plotting points on a graph (Quick Review)

2. Solving for a variable in an equation (Quick Review)

3. Understanding slope (rise over run) (Quick Review)

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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:

• 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

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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.)

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5. Peer Practice (We Do – Students Work Together)

With a partner, solve:

1. Problem: A line passes through (0, -4) with slope 1/2. What's the equation?

   • Hint: Start with y = mx + b and find b.

2. Problem: Find the equation for points (-1, 6) and (1, 2).

   • Hint: First, find slope (m), then b.

3. Problem: Rewrite 3x-y = 5 in slope-intercept form.

   • Hint: Isolate y!

💬 Discuss: Did you and your partner get the same answer?

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6. Independent Practice (You Do – Students Try Alone)

Try these on your own (solutions at the end):

1. Find the equation of a line with slope 4 and y-intercept -7.

2. A line passes through (3, 10) and (5, 16). Find its equation.

3. Rewrite 5x + 2y = 10 in slope-intercept form.

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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).

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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.

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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!

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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!

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Problems on the Slope of a Line

1. Find the slope of the line passing through (2, 5) and (4, 9).
   Solution:
   $m=\frac{9-5}{4-2}=\frac{4}{2}=$ Click to answer

2. Determine the slope of the line passing through (-1, 3) and (5, -7).
   Solution:
   $m=\frac{-7-3}{5-(-1)}=\frac{-10}{6}=$ Click to answer

3. A line has a slope of $\frac{3}{4}$ and passes through (6, -2). Find another point on the line.
   Solution:
   Using $m=\frac{y_2-y_1}{x_2-x_1}$, let $x_2=10$:
   $\frac{3}{4}=\frac{y_2-(-2)}{10-6}$→ $y_2=\; Click\; to\; answer<span\; style="font-family:\; "Times\; New\; Roman";\; font-size:\; medium;\; font-weight:\; 700;"><br></span>$

4. Find the slope of the line $2x+3y=6$.
   Solution:
   Rewrite in slope-intercept form:
   $3y=-2x+6$ → $y=\frac{-2}{3}x+2$ → Slope $m=$Click to answer

5. A line passes through (0, 4) and has a slope of -5. Write its equation.
   Solution:
   Using $y=mx+b$, $b=4$:
   Equation: $y=$ Click to answer

6. Two points on a line are (3, -1) and (7, k). If the slope is 2, find k.
   Solution:
   $2=\frac{k-(-1)}{7-3}$ → $2=\frac{k+1}{4}$ → $k=$ Click to answer

7. Find the slope of a line perpendicular to $y=\frac{2}{5}x-3$.
   Solution:
   Perpendicular slope = negative reciprocal:
   $m_{\perp }=$ Click to answer

8. If a line is horizontal and passes through (4, -2), what is its slope?
   Solution:
   Horizontal lines have slope $m=$Click to answer

9. A line has an undefined slope and passes through (5, -3). Write its equation.
   Solution:
   Lines with undefined slope are vertical:
   Equation: $x=$Click to answer

10. The slope between (a, 8) and (-3, 2) is -2. Find the value of a.
    Solution:
    $-2=\frac{2-8}{-3-a}$ → $-2=\frac{-6}{-3-a}$ → $a=$ Click to answer

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Practice Problems with Partial Solutions

1. Find the slope of the line passing through (2, 5) and (4, 9).
   Solution:
   $m=\frac{9-5}{4-2}=\frac{4}{2}=2$

2. Determine the slope of the line passing through (-1, 3) and (5, -7).
   Solution:
   $m=\frac{-7-3}{5-(-1)}=\frac{-10}{6}=-\frac{5}{3}$

3. A line has a slope of $\frac{3}{4}$ and passes through (6, -2). Find another point on the line.
   Solution:
   Using $m=\frac{y_2-y_1}{x_2-x_1}$, let $x_2=10$:
   $\frac{3}{4}=\frac{y_2-(-2)}{10-6}$ → $\frac{3}{4}=\frac{y_2+2}{4}$→ $y_2+2=3$ → $y_2=1$
   Another point: $(10,1)$
   

4. Find the slope of the line $2x+3y=6$.
   Solution:
   Rewrite in slope-intercept form:
   $3y=-2x+6$ → $y=-\frac{2}{3}x+2$
   Slope $m=-\frac{2}{3}$

5. A line passes through (0, 4) and has a slope of -5. Write its equation.
   Solution:
   Using $y=mx+b$, $b=4$:
   Equation: $y=-5x+4$
   

6. Two points on a line are (3, -1) and (7, k). If the slope is 2, find k.
   Solution:
   $2=\frac{k-(-1)}{7-3}$ → $2=\frac{k+1}{4}$ → $k+1=8$ → $k=7$.

7. Find the slope of a line perpendicular to $y=\frac{2}{5}x-3$.
   Solution:
   Perpendicular slope = negative reciprocal:
   $m_{\perp }=-\frac{5}{2}$

8. If a line is horizontal and passes through (4, -2), what is its slope?
   Solution:
   Horizontal lines have slope $m=0$
   

9. A line has an undefined slope and passes through (5, -3). Write its equation.
   Solution:
   Lines with undefined slope are vertical:
   Equation: $x=5$
   

10. The slope between (a, 8) and (-3, 2) is -2. Find the value of a.
    Solution:
    $-2=\frac{2-8}{-3-a}$ → $-2=\frac{-6}{-3-a}$
    −2(−3−a)=−6 → $6+2a=-6$
    

11. Further Resources

📚 Learn More:

• Slope of a Line

• Equation of a Line

Next Lesson: Point-Slope Form & Standard Form 🚀