Solved Exercises on Understanding Linear Relationships in Grade 8

Master linear relationships: slope-intercept form, point-slope form, graphing, identifying linear vs non-linear relationships, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Slope-Intercept Form
Exercise 1
Identify the slope and y-intercept of:
\(y = 3x + 2\)
Definition:

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

Slope-Intercept Method:

Slope-Intercept Form: \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept

  1. Identify the coefficient of \(x\) (slope)
  2. Identify the constant term (y-intercept)
  3. Interpret what these values mean
Original Equation
\(y = 3x + 2\)
Identify Slope
\(m = 3\)
Identify Y-Intercept
\(b = 2\)
Step 1: Recognize the form

The equation is in slope-intercept form: \(y = mx + b\)

Step 2: Identify the slope

The slope is the coefficient of \(x\): \(m = 3\)

Step 3: Identify the y-intercept

The y-intercept is the constant term: \(b = 2\)

Step 4: Interpret the meaning

Slope = 3 means the line rises 3 units for every 1 unit it runs right

Y-intercept = 2 means the line crosses the y-axis at point (0, 2)

Step 5: Verify by graphing

Starting at (0, 2), move right 1 unit and up 3 units to get (1, 5)

Plotting these points and drawing a line confirms the relationship

Slope: \(m = 3\), Y-intercept: \(b = 2\)
Final answer:

Slope: \(m = 3\), Y-intercept: \(b = 2\)

Applied rules:

• Slope-Intercept Form: \(y = mx + b\) where \(m\) is slope and \(b\) is y-intercept

• Slope Interpretation: Rise over run

• Y-Intercept: Point where line crosses y-axis (x = 0)

2 Point-Slope Form
Exercise 2
Write the equation of a line passing through (2, 5) with slope 4
Definition:

Point-Slope Form: \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a point and \(m\) is the slope

Given Information
Point: (2, 5), Slope: \(m = 4\)
Point-Slope Formula
\(y - y_1 = m(x - x_1)\)
Substitute Values
\(y - 5 = 4(x - 2)\)
Distribute
\(y - 5 = 4x - 8\)
Solve for y
\(y = 4x - 3\)
Step 1: Identify given information

Point: \((x_1, y_1) = (2, 5)\), Slope: \(m = 4\)

Step 2: Write the point-slope formula

\(y - y_1 = m(x - x_1)\)

Step 3: Substitute known values

\(y - 5 = 4(x - 2)\)

Step 4: Distribute the slope

\(y - 5 = 4x - 8\)

Step 5: Solve for y to get slope-intercept form

Add 5 to both sides: \(y = 4x - 8 + 5\)

So: \(y = 4x - 3\)

Step 6: Verify the solution

Check that the point (2, 5) satisfies the equation: \(y = 4(2) - 3 = 8 - 3 = 5\) ✓

\(y = 4x - 3\)
Final answer:

\(y = 4x - 3\)

Applied rules:

• Point-Slope Form: \(y - y_1 = m(x - x_1)\)

• Distributive Property: \(a(b + c) = ab + ac\)

• Equation Manipulation: Solve for desired variable

3 Graphing Linear Equations
Exercise 3
Graph the equation \(y = -2x + 4\) and identify the x-intercept
Definition:

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

Original Equation
\(y = -2x + 4\)
Identify Slope and Y-Intercept
\(m = -2, b = 4\)
Find X-Intercept
Set \(y = 0\): \(0 = -2x + 4\)
Solve for x
\(x = 2\)
Step 1: Identify the y-intercept

From \(y = -2x + 4\), the y-intercept is \(b = 4\), so the point is (0, 4)

Step 2: Use the slope to find another point

Slope is \(-2\), which means rise = \(-2\) and run = \(1\)

Starting from (0, 4), move right 1 unit and down 2 units to get (1, 2)

Step 3: Find the x-intercept

Set \(y = 0\) and solve: \(0 = -2x + 4\)

Add \(2x\) to both sides: \(2x = 4\)

Divide by 2: \(x = 2\)

So x-intercept is (2, 0)

Step 4: Plot the points and draw the line

Plot (0, 4), (1, 2), and (2, 0), then draw a straight line through them

Step 5: Verify the line

Check that the slope is consistent: from (0, 4) to (2, 0), rise = \(-4\) and run = \(2\), so slope = \(\frac{-4}{2} = -2\) ✓

X-intercept: (2, 0)
Final answer:

X-intercept: (2, 0)

Applied rules:

• Graphing Method: Use y-intercept and slope to plot points

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

• Slope Interpretation: Rise over run

Linear Relationships Rules and Methods
\(y = mx + b\)
Slope-Intercept Form
Slope-Intercept
\(y = mx + b\)
\(m\) = slope, \(b\) = y-intercept
Point-Slope
\(y - y_1 = m(x - x_1)\)
Given point \((x_1, y_1)\) and slope \(m\)
Standard Form
\(Ax + By = C\)
Where \(A\), \(B\), and \(C\) are integers
Slope Formula
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Given two points \((x_1, y_1)\) and \((x_2, y_2)\)
X-Intercept
Set \(y = 0\) and solve for \(x\)
Point where line crosses x-axis
Y-Intercept
Set \(x = 0\) and solve for \(y\)
Point where line crosses y-axis
Key Concepts: Linear relationships have a constant rate of change (slope). The graph of a linear equation is always a straight line.
Rate of Change: The slope represents how much the dependent variable changes for each unit change in the independent variable.
Tip 1: To find intercepts, set the other variable to zero and solve.
Tip 2: A positive slope means the line rises from left to right; negative slope means it falls.
Tip 3: Horizontal lines have slope 0; vertical lines have undefined slope.
Tip 4: Parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals.
Solution: Exercises 4 to 5
4 Identifying Linear vs Non-Linear
Exercise 4
Determine if each relationship is linear or non-linear:
a) \(y = 3x^2 + 2\)
b) \(y = 4x - 1\)
c) \(xy = 6\)
Definition:

Linear Relationship: A relationship where the highest power of the variable is 1, forming a straight line when graphed

Relationship a
\(y = 3x^2 + 2\)
Highest Power
\(x^2\) (power = 2)
Type
Non-linear
Relationship b
\(y = 4x - 1\)
Highest Power
\(x^1\) (power = 1)
Type
Linear
Relationship c
\(xy = 6\)
Rearrange
\(y = \frac{6}{x}\)
Type
Non-linear
Step 1: Analyze relationship a: \(y = 3x^2 + 2\)

The highest power of \(x\) is 2 (quadratic), so this is non-linear.

Step 2: Analyze relationship b: \(y = 4x - 1\)

The highest power of \(x\) is 1, so this is linear.

Step 3: Analyze relationship c: \(xy = 6\)

When solved for \(y\): \(y = \frac{6}{x}\), which has \(x\) in the denominator, making it non-linear.

Step 4: Identify key characteristics

Linear relationships have variables raised to the first power only, possibly with coefficients and constants.

Non-linear relationships have variables raised to powers other than 1, or variables in denominators.

a) Non-linear, b) Linear, c) Non-linear
Final answer:

a) Non-linear, b) Linear, c) Non-linear

Applied rules:

• Linear Criterion: Highest power of variable is 1

• Non-Linear Indicators: Powers greater than 1, variables in denominators, products of variables

• Graph Shape: Linear = straight line, Non-linear = curved line

5 Real-World Linear Application
Exercise 5
A taxi charges $3 as a base fare plus $2 per mile. Write an equation relating the cost to the distance, and find the cost for a 10-mile trip.
Definition:

Direct Variation: A linear relationship where one quantity varies directly with another

Define Variables
Let \(C\) = cost, \(d\) = distance in miles
Set Up Equation
\(C = 2d + 3\)
Substitute \(d = 10\)
\(C = 2(10) + 3\)
Calculate
\(C = 23\)
Step 1: Identify the variables

Independent variable: distance (\(d\)), Dependent variable: cost (\(C\))

Step 2: Identify the rate of change

Cost increases by $2 for each mile: rate = $2 per mile

Step 3: Identify the initial value

Base fare when distance = 0: $3

Step 4: Write the linear equation

Cost = (rate per mile) × (distance) + (base fare)

So: \(C = 2d + 3\)

Step 5: Solve for the specific case

For a 10-mile trip: \(C = 2(10) + 3 = 20 + 3 = 23\)

\(C = 2d + 3\), Cost for 10 miles = $23
Final answer:

Equation: \(C = 2d + 3\), Cost for 10 miles = $23

Applied rules:

• Linear Model: \(y = mx + b\) where \(m\) is rate and \(b\) is initial value

• Variable Identification: Determine independent and dependent variables

• Real-World Interpretation: Connect mathematical concepts to practical scenarios

Complete Guide: Linear Relationships, Rules, Methods, and Applications
\(y = mx + b\)
Slope-Intercept Form
Key definitions:

Linear Relationship: A relationship between two variables that can be represented by a straight line on a coordinate plane

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

Y-Intercept: The point where a line crosses the y-axis (where x = 0)

X-Intercept: The point where a line crosses the x-axis (where y = 0)

Complete methodology:
  1. Identify the form: Determine if the equation is in slope-intercept, point-slope, or standard form
  2. Extract parameters: Identify slope and intercepts from the equation
  3. Graph the line: Use slope and intercepts to plot the line
  4. Interpret meaning: Understand what the slope and intercepts represent in context
  5. Verify: Check that points satisfy the equation
Tip 1: Always look for the highest power of variables to determine if a relationship is linear.
Tip 2: The slope tells you how much the dependent variable changes for each unit change in the independent variable.
Tip 3: Use the y-intercept as a starting point when graphing, then use the slope to find additional points.
Tip 4: In real-world problems, the y-intercept often represents an initial value or fixed cost.
Common errors: Confusing slope and y-intercept, misidentifying linear vs non-linear relationships, incorrectly calculating slope from two points.
Real-World Applications: Linear relationships model many real-life situations like pricing, distance-speed-time, temperature conversion, and business scenarios.
Essential rules to memorize:

• Slope-Intercept Form: \(y = mx + b\)

• Point-Slope Form: \(y - y_1 = m(x - x_1)\)

• Standard Form: \(Ax + By = C\)

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

• Linear Characteristic: Variables have exponents of 1

Exercise with Visualization: Linear Relationships
Exercise 6: Linear Function Comparison
Consider the following linear functions:
\(f_1(x) = 2x + 1\)
\(f_2(x) = -x + 3\)
\(f_3(x) = 0.5x - 2\)

Analysis: The chart shows how different linear functions have different slopes and y-intercepts.

  • \(f_1(x) = 2x + 1\) (positive slope: rises)
  • \(f_2(x) = -x + 3\) (negative slope: falls)
  • \(f_3(x) = 0.5x - 2\) (gentle positive slope)

Questions & Answers

Question: How do I know if a relationship is linear or not? What makes a relationship linear?

Answer: A relationship is linear if it meets these criteria:

1. Constant Rate of Change: The slope between any two points is the same.

2. Variables to First Power: In the equation, variables are raised only to the power of 1.

3. Graph Forms a Straight Line: When plotted, all points lie on a straight line.

Examples of Linear:

  • \(y = 3x + 2\) (highest power of x is 1)
  • \(2x + 3y = 6\) (standard form with variables to power 1)
  • \(y = -4x\) (direct variation)

Examples of Non-Linear:

  • \(y = x^2 + 3\) (has \(x^2\) term)
  • \(y = \frac{2}{x}\) (variable in denominator)
  • \(y = x^3 - 2x + 1\) (has \(x^3\) term)

Linear relationships have a constant rate of change, meaning for every unit increase in x, y changes by the same amount.

Question: What does the slope actually represent in real life?

Answer: The slope represents the rate of change between two variables:

Examples:

  • Distance-Time Graph: Slope = speed (miles per hour)
  • Cost-Quantity Graph: Slope = price per item ($ per item)
  • Temperature-Time Graph: Slope = rate of heating/cooling (degrees per minute)
  • Population-Time Graph: Slope = growth rate (people per year)

General interpretation: Slope = \(\frac{\text{change in dependent variable}}{\text{change in independent variable}}\)

So if you have a linear equation \(C = 2.5t + 10\) where \(C\) is cost and \(t\) is time, the slope 2.5 means the cost increases by $2.50 for each unit of time.

The slope always tells you how much the output changes for each unit change in the input!

Question: How do I find the equation of a line if I only have two points?

Answer: Use these steps to find the equation of a line through two points:

Step 1: Find the slope

Use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Step 2: Use point-slope form

Choose one of the points and substitute into: \(y - y_1 = m(x - x_1)\)

Step 3: Simplify to slope-intercept form

Solve for y to get: \(y = mx + b\)

Example: Find the equation through points (1, 3) and (4, 9)

  1. Slope: \(m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2\)
  2. Point-slope: \(y - 3 = 2(x - 1)\)
  3. Simplify: \(y - 3 = 2x - 2\), so \(y = 2x + 1\)

The equation is \(y = 2x + 1\).

Question: What happens if the slope is 0 or undefined?

Answer: Special cases of slope:

Zero Slope:

  • Occurs when the line is horizontal
  • Equation form: \(y = b\) (constant)
  • Example: \(y = 5\) (line passes through (0, 5) and is flat)
  • The y-value never changes regardless of x-value

Undefined Slope:

  • Occurs when the line is vertical
  • Equation form: \(x = a\) (constant)
  • Example: \(x = 3\) (line passes through (3, 0) and goes up/down)
  • The x-value never changes regardless of y-value

These are special linear relationships, but vertical lines are not functions (they fail the vertical line test).

Question: How do I know which form of the equation to use?

Answer: Choose the form based on the information you have:

Slope-Intercept Form (\(y = mx + b\)):

  • Use when you know the slope and y-intercept
  • Best for graphing quickly
  • Best for interpreting real-world meaning

Point-Slope Form (\(y - y_1 = m(x - x_1)\)):

  • Use when you know a point and the slope
  • Use when you know two points (find slope first)

Standard Form (\(Ax + By = C\)):

  • Use when working with integer coefficients
  • Common in systems of equations
  • Good for finding intercepts

All forms represent the same line, so you can convert between them as needed!