Solved Exercises on Understanding Linear Relationships in Grade 7

Master understanding linear relationships: slope, intercepts, equations, tables, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Finding Slope from Points
Exercise 1
Find the slope of the line passing through points (2, 3) and (5, 9).
Definition:

Slope: The measure of steepness of a line, calculated as rise over run.

Slope formula:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\) where (x₁, y₁) and (x₂, y₂) are points on the line.

Given Points
(2, 3) and (5, 9)
Substitute into formula
m = (9 - 3)/(5 - 2)
Calculate
m = 6/3 = 2
Step 1: Identify the coordinates

Point 1: (x₁, y₁) = (2, 3)

Point 2: (x₂, y₂) = (5, 9)

Step 2: Apply the slope formula

m = (y₂ - y₁)/(x₂ - x₁)

m = (9 - 3)/(5 - 2)

Step 3: Calculate the differences

Vertical change (rise): 9 - 3 = 6

Horizontal change (run): 5 - 2 = 3

Step 4: Compute the slope

m = 6/3 = 2

m = 2
Final answer:

The slope of the line is 2.

Applied rules:

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

• Order matters: Consistent assignment of coordinates

• Positive slope: Line rises from left to right

2 Identifying Linear Relationships
Exercise 2
Determine if the table below represents a linear relationship:
x: 1, 2, 3, 4
y: 5, 8, 11, 14
Definition:

Linear relationship: A relationship where the rate of change between variables is constant.

Table Values
x: 1, 2, 3, 4; y: 5, 8, 11, 14
Calculate Differences
Δx = 1, Δy = 3
Constant Rate
Yes, slope = 3
Step 1: Calculate consecutive differences in x

From x = 1 to x = 2: 2 - 1 = 1

From x = 2 to x = 3: 3 - 2 = 1

From x = 3 to x = 4: 4 - 3 = 1

Step 2: Calculate consecutive differences in y

From y = 5 to y = 8: 8 - 5 = 3

From y = 8 to y = 11: 11 - 8 = 3

From y = 11 to y = 14: 14 - 11 = 3

Step 3: Check for constant rate of change

Rate of change = Δy/Δx = 3/1 = 3 for all pairs

Step 4: Conclusion

Since the rate of change is constant, this is a linear relationship.

Yes, this is a linear relationship with slope = 3
Final answer:

Yes, the table represents a linear relationship with a constant rate of change of 3.

Applied rules:

• Constant rate: For linear relationships, Δy/Δx must be constant

• Difference test: Calculate consecutive differences in both x and y

• Linearity: Equal intervals in x correspond to equal intervals in y

3 Writing Linear Equations
Exercise 3
Write the equation of a line with slope 4 and y-intercept -2.
Definition:

Slope-intercept form: y = mx + b, where m is slope and b is y-intercept.

Given Information
m = 4, b = -2
Substitute into form
y = mx + b
Final Equation
y = 4x - 2
Step 1: Identify the form

The slope-intercept form is: y = mx + b

Step 2: Identify given values

Slope (m) = 4

y-intercept (b) = -2

Step 3: Substitute values

y = mx + b

y = 4x + (-2)

y = 4x - 2

Step 4: Verify the equation

When x = 0: y = 4(0) - 2 = -2 (y-intercept confirmed)

Slope is 4 (as given)

y = 4x - 2
Final answer:

The equation of the line is y = 4x - 2.

Applied rules:

• Slope-intercept form: y = mx + b

• Direct substitution: Replace m and b with given values

• Y-intercept: Value of y when x = 0

Linear Relationships Rules and Methods
y = mx + b
Slope-Intercept Form
Slope Formula
m = (y₂ - y₁)/(x₂ - x₁)
Rate of change between two points
Point-Slope Form
y - y₁ = m(x - x₁)
Using point and slope
Standard Form
Ax + By = C
General form of linear equation
Key definitions:

Linear relationship: A relationship between two variables that forms a straight line when graphed.

Slope (m): The steepness of a line, calculated as the ratio of vertical change to horizontal change.

Y-intercept (b): The point where the line crosses the y-axis (when x = 0).

X-intercept: The point where the line crosses the x-axis (when y = 0).

Rate of change: How much one variable changes relative to another variable.

Linear relationship analysis:
  1. Identify variables: Determine independent and dependent variables
  2. Calculate slope: Find the rate of change between points
  3. Find intercepts: Locate where the line crosses axes
  4. Write equation: Express relationship in mathematical form
  5. Verify linearity: Confirm constant rate of change
Tip 1: Positive slope rises from left to right; negative slope falls from left to right.
Tip 2: Horizontal lines have zero slope; vertical lines have undefined slope.
Tip 3: Always verify linearity by checking for constant rate of change.
Tip 4: The y-intercept occurs when x = 0; the x-intercept occurs when y = 0.
Common errors: Mixing up coordinates in slope formula, forgetting to check for constant rate of change, misidentifying intercepts, confusing slope and intercept values.
Exam preparation: Practice finding slopes from graphs and tables, master all forms of linear equations, understand real-world applications, work with negative slopes and intercepts.
Essential rules:

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

• Slope-intercept form: y = mx + b

• Point-slope form: y - y₁ = m(x - x₁)

• Linear relationships have constant rate of change

• Y-intercept is value of y when x = 0

Solution: Exercises 4 to 5
4 Intercepts and Graphing
Exercise 4
Find the x-intercept and y-intercept of the line 2x + 3y = 12, then graph the line.
Definition:

Intercepts: Points where a line crosses the coordinate axes.

Original Equation
2x + 3y = 12
Y-intercept (x = 0)
(0, 4)
X-intercept (y = 0)
(6, 0)
Step 1: Find y-intercept

Set x = 0: 2(0) + 3y = 12

3y = 12

y = 4

y-intercept: (0, 4)

Step 2: Find x-intercept

Set y = 0: 2x + 3(0) = 12

2x = 12

x = 6

x-intercept: (6, 0)

Step 3: Plot the intercepts

Plot points (0, 4) and (6, 0)

Step 4: Draw the line

Connect the two points with a straight line

X-intercept: (6, 0), Y-intercept: (0, 4)
Final answer:

X-intercept: (6, 0), Y-intercept: (0, 4). The line passes through these points.

Applied rules:

• Y-intercept: Set x = 0 and solve for y

• X-intercept: Set y = 0 and solve for x

• Graphing: Two points determine a unique line

5 Real-World Application
Exercise 5
A company charges $25 per hour for consulting plus a $50 setup fee. Write an equation representing the total cost as a function of hours worked, and interpret the slope and y-intercept.
Definition:

Real-world applications: Linear relationships model many practical situations with constant rates.

Define Variables
Let x = hours, y = total cost
Write Equation
y = 25x + 50
Interpret Components
Slope = 25, Y-int = 50
Step 1: Define variables

Let x = number of hours worked

Let y = total cost in dollars

Step 2: Identify components

Hourly rate: $25 per hour

Setup fee: $50 (fixed cost)

Step 3: Write the equation

Total cost = (hourly rate × hours) + setup fee

y = 25x + 50

Step 4: Interpret slope

Slope = 25: Cost increases by $25 for each additional hour

Step 5: Interpret y-intercept

Y-intercept = 50: Cost is $50 when x = 0 (setup fee)

y = 25x + 50; slope = 25, y-intercept = 50
Final answer:

Equation: y = 25x + 50. The slope (25) represents the hourly rate, and the y-intercept (50) represents the setup fee.

Applied rules:

• Modeling: Translate real-world situations into mathematical equations

• Slope interpretation: Represents the rate of change in context

• Y-intercept interpretation: Represents the initial value or fixed cost

Detailed Linear Relationships Guide
y = mx + b
Slope-Intercept Form
Key definitions:

Linear relationship: A relationship between two variables where one variable changes at a constant rate with respect to the other. When graphed, it forms a straight line. The general form is y = mx + b.

Slope (m): The measure of the steepness of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. It represents the rate of change.

Y-intercept (b): The point where the line crosses the y-axis. This is the value of y when x = 0.

X-intercept: The point where the line crosses the x-axis. This is the value of x when y = 0.

Rate of change: How much the dependent variable (y) changes for each unit change in the independent variable (x).

Constant rate: For linear relationships, the rate of change between variables remains the same throughout the relationship.

Complete linear relationship analysis:
  1. Identify variables: Determine which variable is independent (x) and which is dependent (y)
  2. Calculate slope: Use the formula m = (y₂ - y₁)/(x₂ - x₁) to find the rate of change
  3. Find intercepts: Calculate where the line crosses the x and y axes
  4. Write equation: Express the relationship in slope-intercept form (y = mx + b)
  5. Verify linearity: Confirm that the rate of change is constant across all points
  6. Interpret results: Understand the meaning of slope and intercepts in context
Tip 1: To remember slope, think "rise over run" - how much up/down divided by how much left/right.
Tip 2: Positive slope means the line goes up as you move right; negative slope means it goes down.
Tip 3: To find intercepts, set one variable to zero and solve for the other.
Tip 4: Always verify your linear relationship by checking that the rate of change is constant.
Common errors: Mixing up coordinates in the slope formula, confusing slope and y-intercept values, forgetting that horizontal lines have zero slope, assuming all relationships are linear without verification.
Applications: Distance-speed-time problems, cost calculations, conversion formulas, growth models, scientific measurements, business analysis, and economic relationships.
Essential linear relationship rules:

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

• Slope-intercept form: y = mx + b

• Point-slope form: y - y₁ = m(x - x₁)

• Standard form: Ax + By = C

• Linear relationships have constant rate of change

• Y-intercept occurs when x = 0

• X-intercept occurs when y = 0

Linear Relationships Guide

📊
Slope Formula

m = (y₂ - y₁)/(x₂ - x₁)

Rate of change

Steepness measure

Forms

y = mx + b

Slope-intercept

m = slope, b = y-int

Interpretation

Slope = rate of change

Y-intercept = starting value

Constant rate = linear

Verification

Check constant rate

Equal intervals in x

Equal intervals in y

Questions & Answers

Question: What does the slope actually tell me about a linear relationship?

Answer: The slope tells you the rate of change in the relationship - how much the dependent variable (y) changes for each unit change in the independent variable (x).

Examples:

  • If slope = 3: For every 1 unit increase in x, y increases by 3 units
  • If slope = -2: For every 1 unit increase in x, y decreases by 2 units
  • If slope = 0.5: For every 1 unit increase in x, y increases by 0.5 units

In real-world contexts:

  • Cost vs. time: slope = cost per unit time
  • Distance vs. time: slope = speed
  • Temperature vs. altitude: slope = rate of temperature change

The slope is the "speed" or "rate" of the relationship!

Question: How do I know if a relationship is linear from a table of values?

Answer: A relationship is linear if the rate of change (slope) is constant between any two points in the table.

Steps to check:

  1. Calculate the change in x (Δx) between consecutive rows
  2. Calculate the change in y (Δy) between consecutive rows
  3. Calculate the ratio Δy/Δx for each pair of consecutive points
  4. If all ratios are equal, the relationship is linear

Example: Table with x: 1, 2, 3, 4 and y: 5, 8, 11, 14

  • From (1,5) to (2,8): Δx = 1, Δy = 3, ratio = 3/1 = 3
  • From (2,8) to (3,11): Δx = 1, Δy = 3, ratio = 3/1 = 3
  • From (3,11) to (4,14): Δx = 1, Δy = 3, ratio = 3/1 = 3

Since all ratios equal 3, this is a linear relationship with slope = 3.

Question: What's the difference between slope-intercept form and standard form of a linear equation?

Answer: The main differences are in structure and purpose:

Slope-intercept form (y = mx + b):

  • Explicitly shows slope (m) and y-intercept (b)
  • Easy to graph and interpret
  • Directly reveals rate of change and starting point
  • Best for analysis and understanding

Standard form (Ax + By = C):

  • All variables on one side, constant on the other
  • Coefficients are typically integers
  • Good for algebraic manipulation
  • Preferred for systems of equations

Conversion:

  • Standard to slope-intercept: Solve for y
  • Slope-intercept to standard: Rearrange to get x and y terms on one side

Both represent the same line, just in different formats for different purposes!