Solved Exercises on Slope Introduction in Integrated Math 1

Master slope: calculating slope, interpreting steepness, and applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Slope Calculation
Exercise 1
Find the slope of the line passing through points A(2, 3) and B(6, 7). Show your work using the slope formula.
Definition:

Slope: The measure of steepness of a line, calculated as the ratio of vertical change to horizontal change: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Slope Formula Method:
  1. Identify two points on the line: \((x_1, y_1)\) and \((x_2, y_2)\)
  2. Apply the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
  3. Substitute the coordinates into the formula
  4. Calculate the result and simplify if needed
Points
A(2,3), B(6,7)
Formula
\(m = \frac{7-3}{6-2}\)
Result
\(m = \frac{4}{4} = 1\)
Step 1: Identify the coordinates

Point A: \((x_1, y_1) = (2, 3)\)

Point B: \((x_2, y_2) = (6, 7)\)

Step 2: Apply the slope formula

\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{6 - 2}\)

Step 3: Calculate the differences

\(m = \frac{4}{4} = 1\)

Step 4: Interpret the result

The slope is positive 1, indicating the line rises as it moves from left to right

Slope = 1
Final answer:

The slope of the line is \(m = 1\). This means for every 1 unit moved horizontally, the line rises 1 unit vertically.

Applied rules:

• Slope Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

• Positive Slope: Line rises from left to right

• Order Independence: \((x_1, y_1)\) and \((x_2, y_2)\) can be swapped as long as consistent

2 Negative Slope
Exercise 2
Calculate the slope of the line passing through points C(-1, 5) and D(3, -3). Describe what this slope tells us about the line.
Definition:

Negative Slope: A slope with a negative value indicates a line that falls as it moves from left to right

Points
C(-1,5), D(3,-3)
Formula
\(m = \frac{-3-5}{3-(-1)}\)
Result
\(m = \frac{-8}{4} = -2\)
Step 1: Identify the coordinates

Point C: \((x_1, y_1) = (-1, 5)\)

Point D: \((x_2, y_2) = (3, -3)\)

Step 2: Apply the slope formula

\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 5}{3 - (-1)}\)

Step 3: Calculate the differences

\(m = \frac{-8}{4} = -2\)

Step 4: Interpret the result

The slope is negative 2, indicating the line falls 2 units for every 1 unit moved horizontally

Slope = -2
Final answer:

The slope of the line is \(m = -2\). This means the line falls 2 units for every 1 unit moved to the right.

Applied rules:

• Negative Slope: Line falls from left to right

• Steepness: Absolute value indicates steepness (higher absolute value = steeper)

• Direction: Negative slope means decreasing trend

3 Zero and Undefined Slope
Exercise 3
Find the slopes of the lines passing through: a) E(2, 4) and F(6, 4), and b) G(3, 1) and H(3, 7). Explain what each type of slope represents.
Definition:

Zero Slope: A horizontal line with slope \(m = 0\). Undefined Slope: A vertical line with no defined slope (denominator = 0).

Case a
E(2,4), F(6,4)
Case b
G(3,1), H(3,7)
Results
m=0, m=undefined
Step 1: Calculate slope for E(2,4) and F(6,4)

\(m = \frac{4 - 4}{6 - 2} = \frac{0}{4} = 0\)

This is a horizontal line (zero slope)

Step 2: Calculate slope for G(3,1) and H(3,7)

\(m = \frac{7 - 1}{3 - 3} = \frac{6}{0}\)

This is undefined (vertical line)

Step 3: Interpret the results

Case a: Horizontal line - no vertical change, y-values are constant

Case b: Vertical line - no horizontal change, x-values are constant

Step 4: Generalize the patterns

Horizontal lines: form y = constant, slope = 0

Vertical lines: form x = constant, slope = undefined

a) m = 0, b) m = undefined
Final answer:

a) The slope is 0 (horizontal line). b) The slope is undefined (vertical line).

Applied rules:

• Zero Slope: Horizontal line, y-values constant

• Undefined Slope: Vertical line, x-values constant

• Division by Zero: Creates undefined result

Slope Rules and Properties
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Slope Formula
Positive Slope
m > 0
Line rises right
Negative Slope
m < 0
Line falls right
Zero Slope
m = 0
Horizontal line
Key definitions:

Slope: The ratio of vertical change (rise) to horizontal change (run) between any two points on a line

Rise: The change in y-coordinates (vertical change)

Run: The change in x-coordinates (horizontal change)

Slope Calculation Methodology:
  1. Identify Points: Locate two distinct points on the line
  2. Label Coordinates: Assign \((x_1, y_1)\) and \((x_2, y_2)\) to the points
  3. Apply Formula: Use \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
  4. Calculate: Compute the rise and run
  5. Interpret: Determine the line's direction and steepness
Tip 1: Remember "rise over run" - vertical change over horizontal change.
Tip 2: Positive slope goes uphill, negative slope goes downhill.
Tip 3: Be careful with negative coordinates when subtracting.
Tip 4: Same result regardless of which point you call (x₁,y₁).
Common Mistakes: Incorrect order of subtraction, sign errors with negatives, confusing rise and run.
Memorization Tip: "Rise over Run" = Vertical change over Horizontal change.
Solution: Exercises 4 to 5
4 Comparing Slopes
Exercise 4
Find the slopes of the lines passing through: a) (1, 2) and (4, 8), b) (-2, 3) and (2, 7), c) (0, 0) and (5, 1). Order them from least steep to most steep.
Definition:

Steepness: Determined by the absolute value of the slope. Higher absolute value = steeper line

Line a
\(m = \frac{8-2}{4-1} = 2\)
Line b
\(m = \frac{7-3}{2-(-2)} = 1\)
Line c
\(m = \frac{1-0}{5-0} = 0.2\)
Step 1: Calculate slope for line a (1,2) and (4,8)

\(m_a = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2\)

Step 2: Calculate slope for line b (-2,3) and (2,7)

\(m_b = \frac{7 - 3}{2 - (-2)} = \frac{4}{4} = 1\)

Step 3: Calculate slope for line c (0,0) and (5,1)

\(m_c = \frac{1 - 0}{5 - 0} = \frac{1}{5} = 0.2\)

Step 4: Order by steepness (absolute value)

\(|m_c| = 0.2\), \(|m_b| = 1\), \(|m_a| = 2\)

Order from least to most steep: c, b, a

Slopes: 0.2, 1, 2
Final answer:

Line c: \(m = 0.2\), Line b: \(m = 1\), Line a: \(m = 2\). Ordered from least to most steep: c, b, a.

Applied rules:

• Steepness Comparison: Compare absolute values of slopes

• Slope Calculation: Apply formula consistently

• Ordering: Arrange by absolute value magnitude

5 Real-World Application
Exercise 5
A car rental company charges $30 per day plus a $50 initial fee. If we plot cost vs. days rented, what is the slope of the resulting line? Interpret this slope in context.
Definition:

Rate of Change: In real-world contexts, slope often represents the rate at which one quantity changes with respect to another

Cost Function
\(C = 30d + 50\)
Slope Form
\(y = mx + b\)
Result
\(m = 30\)
Step 1: Identify the relationship

Total cost = daily rate × days + initial fee

\(C = 30d + 50\)

Step 2: Identify the slope-intercept form

The equation is in the form \(y = mx + b\) where \(m\) is the slope

Here: \(C = 30d + 50\), so \(m = 30\)

Step 3: Calculate slope using two points

Point 1: Day 0, Cost $50 → (0, 50)

Point 2: Day 1, Cost $80 → (1, 80)

\(m = \frac{80 - 50}{1 - 0} = \frac{30}{1} = 30\)

Step 4: Interpret in context

The slope of 30 means the cost increases by $30 for each additional day rented

Slope = 30 ($/day)
Final answer:

The slope is 30, which means the cost increases by $30 per day. This represents the daily rental rate.

Applied rules:

• Real-World Interpretation: Slope represents rate of change

• Linear Modeling: Identify dependent and independent variables

• Rate of Change: What happens when the independent variable increases by 1

Slope Summary: Definitions, Rules, and Applications
\(m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\)
Slope Definition
Key definitions:

Slope: The measure of steepness and direction of a line, calculated as the ratio of vertical change to horizontal change.

Positive Slope: Line rises from left to right (m > 0).

Negative Slope: Line falls from left to right (m < 0).

Zero Slope: Horizontal line (m = 0).

Undefined Slope: Vertical line (division by zero).

Complete Slope Calculation Methodology:
  1. Point Selection: Choose any two points on the line
  2. Coordinate Assignment: Label as \((x_1, y_1)\) and \((x_2, y_2)\)
  3. Formula Application: Apply \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
  4. Arithmetic: Calculate rise and run carefully
  5. Interpretation: Determine sign and magnitude meaning
  6. Verification: Check with alternative points if available
Tip 1: Rise over run: change in y divided by change in x.
Tip 2: Positive slope = uphill from left to right.
Tip 3: Negative slope = downhill from left to right.
Tip 4: Steeper lines have slopes with larger absolute values.
Common Errors: Subtracting coordinates in wrong order, sign mistakes with negatives, confusing rise and run.
Exam Preparation: Practice with various point combinations, understand real-world interpretations, master sign conventions.
Essential Rules and Properties:

• Slope Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

• Positive Slope: Line rises (m > 0)

• Negative Slope: Line falls (m < 0)

• Zero Slope: Horizontal line (m = 0)

• Undefined Slope: Vertical line (division by zero)

Questions & Answers

Question: Does it matter which point I call (x₁, y₁) and which I call (x₂, y₂)? Will I get a different answer?

Answer: No, you'll get the same answer regardless of which point you assign as (x₁, y₁) and which as (x₂, y₂). Here's why:

If you have points A(x₁, y₁) and B(x₂, y₂), the slope is:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

If you reverse the assignment:

\(m = \frac{y_1 - y_2}{x_1 - x_2} = \frac{-(y_2 - y_1)}{-(x_2 - x_1)} = \frac{y_2 - y_1}{x_2 - x_1}\)

The negatives cancel out, giving the same result. Just be consistent with which point you use for the numerator and denominator.

Question: How do I know if a line is steep just by looking at the slope number?

Answer: The steepness of a line is determined by the absolute value of the slope:

  • |m| < 1: Gentle slope (less steep)
  • |m| = 1: Moderate slope (45-degree angle)
  • |m| > 1: Steep slope (steeper than 45 degrees)
  • |m| = 0: Completely flat (horizontal line)

For example, a slope of -5 is steeper than a slope of 2 because |-5| = 5 > |2| = 2.

Question: Why is the slope of a vertical line undefined rather than zero?

Answer: When calculating the slope of a vertical line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), the x-coordinates are identical for any two points on the line.

For example, if we have points (3, 1) and (3, 5) on a vertical line:

\(m = \frac{5 - 1}{3 - 3} = \frac{4}{0}\)

Since division by zero is undefined in mathematics, the slope of a vertical line is undefined. This makes sense intuitively because a vertical line rises infinitely steeply - it doesn't have a defined "rate" of change.