Understand Slope

Grade 8 · algebra · 89 practice problems · read aloud

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Understanding Slope

📈 What is Slope?

Slope is a number that describes the steepness and direction of a line. It tells us how much a line goes up (or down) as it moves to the right. We use it everywhere, from calculating the pitch of a roof to understanding how fast something is moving (speed!).

🛠️ How to Find Slope

The most common way is using the formula with two points: (x₁, y₁) and (x₂, y₂).

Formula: Slope (m) = (y₂ - y₁) / (x₂ - x₁)

This is often remembered as "rise over run".

  1. Step 1: Choose two points on the line.
  2. Step 2: Label them (x₁, y₁) and (x₂, y₂).
  3. Step 3: Subtract the y-values: (y₂ - y₁). This is the rise.
  4. Step 4: Subtract the x-values: (x₂ - x₁). This is the run.
  5. Step 5: Divide the rise by the run.

✏️ Worked Examples

Example 1: Find the slope between (1, 2) and (4, 8).

  1. Points: (x₁, y₁) = (1, 2), (x₂, y₂) = (4, 8)
  2. Rise (y₂ - y₁) = 8 - 2 = 6
  3. Run (x₂ - x₁) = 4 - 1 = 3
  4. Slope = 6 / 3 = 2

The slope is 2. This means the line rises 2 units for every 1 unit it runs.

Example 2: Find the slope between (-3, 5) and (2, -5).

  1. Points: (x₁, y₁) = (-3, 5), (x₂, y₂) = (2, -5)
  2. Rise (y₂ - y₁) = -5 - 5 = -10
  3. Run (x₂ - x₁) = 2 - (-3) = 2 + 3 = 5
  4. Slope = -10 / 5 = -2

The slope is -2. This means the line falls 2 units for every 1 unit it runs.

⚠️ Common Mistakes

  • Mixing up x and y: Always do y₂ - y₁ divided by x₂ - x₁. A good trick is to remember "Y comes before X in the alphabet, but you find Rise (Y) over Run (X)!"
  • Subtracting in the wrong order: If you start with y₂ in the numerator, you MUST start with x₂ in the denominator. Be consistent!
  • Forgetting negative signs: A negative slope is perfectly fine! It just means the line is going downhill from left to right.

💡 Tips & Tricks

  • Memory Aid: "Rise Over Run" – The vertical change comes first, over the horizontal change.
  • Direction Guide:
    • Positive Slope: Line goes ↗ UP from left to right.
    • Negative Slope: Line goes ↘ DOWN from left to right.
    • Zero Slope: Line is flat → (horizontal).
    • Undefined Slope: Line is straight up | (vertical).

🎯 How to Practice

  • Grab graph paper and plot two points. Draw the line between them and physically count the rise and run.
  • Make flashcards with pairs of points on the front and calculate the slope on the back.
  • Look for slopes in the real world! What's the slope of the stairs? The ramp at the library? A hill?
  • Ask a friend for two random numbers and practice the calculation.

Practice problems

6 of the 89, worked through step by step — try them before opening the answer.

1 (3x + 7) = 22

Hint: To solve for the variable, isolate it by performing inverse operations on both sides of the equation.

Show the answer

Answer: 5

  1. The equation is 3x + 7 = 22. We want to isolate the term with x. First, subtract 7 from both sides to remove the constant term on the left. 3x + 7 - 7 = 22 - 7 This simplifies to: 3x = 15.
  2. Now, we have 3x = 15. To solve for x, divide both sides by 3. 3x / 3 = 15 / 3 This gives: x = 5.
  3. Check the solution by substituting x = 5 back into the original equation: 3(5) + 7 = 15 + 7 = 22, which matches the right-hand side of the original equation. Therefore,

We are solving the equation: (3x + 7) = 22. the correct answer is x = 5.

2 (2x - 8)/4 = 3

Hint: To solve for x, first eliminate the fraction by multiplying both sides by the denominator, then isolate the variable term.

Show the answer

Answer: 10

  1. Multiply both sides by 4 to eliminate the fraction: (2x - 8)/4 × 4 = 3 × 4
  2. This simplifies to: 2x - 8 = 12
  3. Add 8 to both sides: 2x - 8 + 8 = 12 + 8
  4. This simplifies to: 2x = 20
  5. Divide both sides by 2: 2x/2 = 20/2
  6. This gives: x = 10

The answer is 10.

3 (2x - 8) ÷ 2 = 5

Hint: Remember to isolate the variable by performing inverse operations in the correct order.

Show the answer

Answer: 9

  1. The equation is (2x - 8) ÷ 2 = 5
  2. Multiply both sides by 2 to eliminate the division: 2x - 8 = 10
  3. Add 8 to both sides to isolate the term with x: 2x = 18
  4. Divide both sides by 2 to solve for x: x = 9

The answer is 9.

4 (2/3) + (5/6) = ?

Hint: To add fractions, they must have the same denominator. Find a common denominator for the fractions before adding.

Show the answer

Answer: 3/2

  1. Identify the problem. We need to add the fractions 2/3 and 5/6.
  2. Check if the denominators are the same. The denominators are 3 and 6, which are not the same. We cannot add them directly.
  3. Find a common denominator. The least common multiple of 3 and 6 is 6. So, the common denominator will be 6.
  4. Rewrite each fraction with denominator 6. The fraction 5/6 already has denominator 6, so it stays as 5/6. For 2/3, we multiply numerator and denominator by 2: (2 × 2)/(3 × 2) = 4/6.
  5. Now add the fractions with the same denominator. 4/6 + 5/6 = (4 + 5)/6 = 9/6.
  6. Simplify the result. Both 9 and 6 can be divided by 3: 9 ÷ 3 = 3, 6 ÷ 3 = 2. So, 9/6 simplifies to 3/2. Final answer: 3/2

5 (3x + 12) ÷ 3 = ?

Hint: When an expression inside parentheses is divided by a number, you can distribute the division to each term inside.

Show the answer

Answer: x + 4

  1. Rewrite the division as a fraction. (3x + 12) ÷ 3 = (3x + 12)/3
  2. Separate the fraction into two terms. We can split the numerator over the denominator like this: (3x + 12)/3 = (3x)/3 + (12)/3
  3. Simplify each fraction separately. (3x)/3 = x because 3/3 = 1, so 1*x = x. (12)/3 = 4 because 12 divided by 3 is 4.
  4. Combine the simplified terms. x + 4 So the final simplified expression is: x + 4

We start with the expression: (3x + 12) ÷ 3

6 (2x + 8) ÷ 2 = 10

Hint: Remember to isolate the variable by reversing the operations in the correct order.

Show the answer

Answer: 6

  1. The equation is (2x + 8) ÷ 2 = 10
  2. Multiply both sides by 2 to eliminate the division: 2x + 8 = 20
  3. Subtract 8 from both sides to isolate the term with x: 2x = 12
  4. Divide both sides by 2 to solve for x: x = 6

The answer is 6.

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