Derive y=mx+b

Grade 8 · algebra · 100 practice problems · read aloud

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Deriving the Equation y = mx + b

📈 What is y = mx + b and Why is it Useful?

The equation y = mx + b is the slope-intercept form of a linear equation. It's a powerful tool because it instantly tells you two key things about a line:

  • Slope (m): How steep the line is. A slope of 2 means for every 1 unit you go right, you go up 2 units.
  • Y-intercept (b): Where the line crosses the y-axis. This is the starting point when x is 0.

We "derive" this equation when we are given other information (like two points or a slope and a point) and need to find the specific equation for our line.

🛠️ Step-by-Step Guide to Finding y = mx + b

If you are given two points, follow these steps:

  1. Find the slope (m): Use the formula m = (y₂ - y₁) / (x₂ - x₁).
  2. Plug m and one point into the equation: You now have y = mx + b, but you know m, x, and y. Plug them in to solve for b.
  3. Solve for b (the y-intercept): This is the missing piece!
  4. Write the final equation: Put your found m and b back into y = mx + b.

🔢 Visual Examples

Example 1: Find the equation of the line through points (1, 2) and (3, 8).

  1. Find m: m = (8 - 2) / (3 - 1) = 6 / 2 = 3
  2. Plug into y = mx + b: Use point (1, 2). 2 = (3)(1) + b
  3. Solve for b: 2 = 3 + b → b = 2 - 3 → b = -1
  4. Final Equation: y = 3x - 1

Example 2: Slope is -2 and the line passes through (4, 1).

  1. m is given: m = -2
  2. Plug into y = mx + b: 1 = (-2)(4) + b
  3. Solve for b: 1 = -8 + b → 1 + 8 = b → b = 9
  4. Final Equation: y = -2x + 9

⚠️ Common Mistakes to Avoid

  • Mixing up x and y in the slope formula: Always do (change in y) / (change in x). A memory trick is "rise over run".
  • Using the wrong point to solve for b: Once you have m, you can use either of the original points to find b. Just be consistent!
  • Forgetting the sign of b: If b is negative, your equation should be y = mx - b, not y = mx + -b.

💡 Tips & Tricks

  • Double-check your slope: Before solving for b, make sure your slope is correct. A simple error here breaks everything.
  • Verify your answer: Plug the other given point into your final equation. If it works, you know you're right!
  • Memory Aid: "Mr. B, the M&Ms are on the X&Y plane!" It's silly, but it helps remember the order: M (slope) then B (y-intercept) with X and Y.

🎯 How to Practice

Master this skill with these practice ideas:

  • Start by creating your own problems. Pick a random m and b, write the equation, then create two points from it. Scramble the points and try to get back to your original equation!
  • Ask a friend for two points and race to see who can find the equation correctly first.
  • Find practice worksheets online that mix up "given two points" and "given slope and a point" problems.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 2(3x - 5) + 4 = 3x + 8

Hint: First distribute any multiplication across terms in parentheses, then combine like terms on each side of the equation.

Show the answer

Answer: x = 6

To solve linear equations with variables on both sides, use inverse operations to isolate the variable. For example, with 2(x+1) = x+5, you would distribute, then move all variable terms to one side and constants to the other.

2 2(3x - 5) + 4 = 3x + 7

Hint: Distribute any coefficients outside parentheses first, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x = 13/3

  1. Distribute the 2** Multiply 2 by each term inside the parentheses: 2 * 3x = 6x 2 * (-5) = -10 So we have: 6x - 10 + 4 = 3x + 7 **
  2. Combine like terms on the left** -10 + 4 = -6 So: 6x - 6 = 3x + 7 **
  3. Move x terms to one side** Subtract 3x from both sides: 6x - 3x - 6 = 7 3x - 6 = 7 **
  4. Move constant terms to the other side** Add 6 to both sides: 3x = 7 + 6 3x = 13 **
  5. Solve for x** Divide both sides by 3: x = 13/3 **Final answer:** x = 13/3

Let's solve the equation step by step. We start with: 2(3x - 5) + 4 = 3x + 7 **

3 2(3x - 5) + 4 = 3x + 12

Hint: First distribute any multiplication across terms in parentheses, then combine like terms on each side of the equation. Move all terms containing the variable to one side and constants to the other.

Show the answer

Answer: 6

  1. Distribute the 2: 2 × 3x = 6x and 2 × (-5) = -10, so we get 6x - 10 + 4 = 3x + 12
  2. Combine like terms on the left: 6x - 6 = 3x + 12
  3. Subtract 3x from both sides: 6x - 3x - 6 = 12, which gives 3x - 6 = 12
  4. Add 6 to both sides: 3x = 18
  5. Divide both sides by 3: x = 6

The answer is 6.

4 (2.4 × 10⁵) × (3.0 × 10⁻³) = ?

Hint: Multiply the coefficients first, then handle the exponents using exponent rules.

Show the answer

Answer: 720

  1. Multiply the coefficients: 2.4 × 3.0 = 7.2
  2. Multiply the powers of 10: 10⁵ × 10⁻³ = 10⁵⁺⁽⁻³⁾ = 10²
  3. Combine the results: 7.2 × 10²
  4. Convert to standard form: 7.2 × 100 = 720

The answer is 720.

5 Points (1, 6) and (6, 31). Find equation in y = mx + b form.

Hint: First find the slope by calculating the change in y divided by the change in x. Then substitute one point into y = mx + b to solve for the y-intercept.

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Answer: y = 5x + 1

  1. Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using (1, 6) and (6, 31): m = (31 - 6) / (6 - 1) = 25 / 5 = 5.
  2. Use point (1, 6) and m = 5 in y = mx + b: 6 = 5(1) + b.
  3. Solve for b: 6 = 5 + b, so b = 6 - 5 = 1.
  4. Write the equation: y = 5x + 1.

The answer is y = 5x + 1.

6 Points (3, 13) and (7, 29). Find equation in y = mx + b form.

Hint: First, find the slope by calculating the change in y over the change in x. Then, substitute one point into y = mx + b to solve for the y-intercept.

Show the answer

Answer: y = 4x + 1

  1. Calculate the slope m using m = (y2 - y1)/(x2 - x1). Using (3, 13) and (7, 29): m = (29 - 13)/(7 - 3) = 16/4 = 4.
  2. Use point (3, 13) and m = 4 in y = mx + b: 13 = 4(3) + b → 13 = 12 + b.
  3. Solve for b: b = 13 - 12 = 1.
  4. Write the equation: y = 4x + 1.
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