Solved Exercises on Linear Equations in Grade 8

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

Solution: Exercises 1 to 3
1 Slope-Intercept Form
Exercise 1
Write the equation of a line with slope 3 and y-intercept -2. Then graph the line and identify its x-intercept.
Definition:

Slope-Intercept Form: The equation y = mx + b, where m is the slope and b is the y-intercept. This form directly shows the slope and where the line crosses the y-axis.

Equation Writing Method:
  1. Identify the slope (m) and y-intercept (b)
  2. Substitute these values into y = mx + b
  3. Simplify if necessary
  4. Identify x-intercept by setting y = 0 and solving for x
Given Values
m = 3, b = -2
Equation
y = 3x - 2
X-intercept
x = 2/3
Step 1: Write the general form

y = mx + b

Step 2: Substitute the known values

m = 3 and b = -2

y = 3x + (-2)

y = 3x - 2

Step 3: Find the x-intercept

Set y = 0: 0 = 3x - 2

2 = 3x

x = 2/3

Step 4: Graph the line

Plot y-intercept (0, -2), then use slope to find another point

From (0, -2), rise 3 and run 1 to get (1, 1)

y = 3x - 2, x-intercept: (2/3, 0)
Final answer:

The equation of the line is y = 3x - 2. The x-intercept is at (2/3, 0).

Applied rules:

• Slope-Intercept Form: y = mx + b

• Y-intercept: Point where x = 0, coordinates (0, b)

• X-intercept: Point where y = 0, found by solving 0 = mx + b

2 Point-Slope Form
Exercise 2
Find the equation of the line passing through points (2, 5) and (4, 9). Write it in slope-intercept form.
Definition:

Point-Slope Form: The equation y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is useful when you know one point and the slope.

Given Points
(2, 5), (4, 9)
Calculated Slope
m = 2
Final Equation
y = 2x + 1
Step 1: Calculate the slope

m = (y₂ - y₁)/(x₂ - x₁) = (9 - 5)/(4 - 2) = 4/2 = 2

Step 2: Use point-slope form with one point

Using point (2, 5): y - 5 = 2(x - 2)

Step 3: Convert to slope-intercept form

y - 5 = 2x - 4

y = 2x - 4 + 5

y = 2x + 1

Step 4: Verify with the other point

Check with (4, 9): y = 2(4) + 1 = 8 + 1 = 9 ✓

y = 2x + 1
Final answer:

The equation of the line is y = 2x + 1.

Applied rules:

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

• Point-Slope Form: y - y₁ = m(x - x₁)

• Conversion: Distribute and solve for y to get slope-intercept form

3 Standard Form
Exercise 3
Convert the equation y = -2x + 6 to standard form Ax + By = C. Then find the x and y intercepts.
Definition:

Standard Form: The equation Ax + By = C, where A, B, and C are integers, and A is typically positive. This form is useful for finding intercepts and solving systems of equations.

Original Equation
y = -2x + 6
Standard Form
2x + y = 6
Intercepts
x-int: (3,0), y-int: (0,6)
Step 1: Move all variables to one side

y = -2x + 6

2x + y = 6

Step 2: Identify coefficients

A = 2, B = 1, C = 6

Step 3: Find y-intercept (set x = 0)

2(0) + y = 6

y = 6

y-intercept: (0, 6)

Step 4: Find x-intercept (set y = 0)

2x + 0 = 6

2x = 6

x = 3

x-intercept: (3, 0)

2x + y = 6, x-int: (3,0), y-int: (0,6)
Final answer:

The standard form is 2x + y = 6. The x-intercept is (3, 0) and the y-intercept is (0, 6).

Applied rules:

• Standard Form: Ax + By = C where A, B, C are integers

• X-intercept: Found by setting y = 0

• Y-intercept: Found by setting x = 0

Rules and methods, laws,...
\(y = mx + b\)
Slope-Intercept Form
Slope-Intercept
y = mx + b
Shows slope and y-intercept
Point-Slope
y - y₁ = m(x - x₁)
Uses point and slope
Standard Form
Ax + By = C
Integer coefficients
Slope Formula
m = (y₂ - y₁)/(x₂ - x₁)
Finding slope between points
Parallel Lines
m₁ = m₂
Same slopes
Perpendicular Lines
m₁ × m₂ = -1
Slopes are negative reciprocals
Slope: The rate of change of y with respect to x, representing steepness and direction
Intercepts: Points where the line crosses the x or y axes
Key definitions:

Linear Equation: An equation whose graph is a straight line. The highest power of variables is 1.

Slope (m): The measure of steepness of a line, calculated as rise over run (change in y/change in x).

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 one quantity changes in relation to another quantity.

Complete methodology:
  1. Identify given information: Determine what's provided (points, slope, intercepts)
  2. Choose appropriate form: Select slope-intercept, point-slope, or standard form
  3. Substitute known values: Insert values into the chosen formula
  4. Solve for unknowns: Find missing values or convert between forms
  5. Verify solution: Check that the equation satisfies given conditions
  6. Graph if needed: Plot the line using intercepts or slope
Tip 1: Always verify your equation by substituting known points.
Tip 2: Positive slope rises left to right, negative slope falls.
Tip 3: Remember that vertical lines have undefined slope.

Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A taxi charges a $3 base fare plus $2 per mile. Write an equation representing the total cost in terms of miles traveled. What would be the cost for a 7-mile trip?
Definition:

Linear Model: A mathematical equation that represents a real-world situation where one quantity changes at a constant rate relative to another.

Variables
C = cost, m = miles
Equation
C = 2m + 3
7-Mile Cost
$17
Step 1: Identify variables and constants

Independent variable: miles (m)

Dependent variable: cost (C)

Fixed cost (y-intercept): $3

Rate (slope): $2 per mile

Step 2: Write the equation

C = (rate per mile) × (miles) + (base fare)

C = 2m + 3

Step 3: Calculate cost for 7 miles

C = 2(7) + 3 = 14 + 3 = $17

C = 2m + 3, Cost for 7 miles = $17
Final answer:

The equation is C = 2m + 3. A 7-mile trip would cost $17.

Applied rules:

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

• Real-World Context: Identify independent and dependent variables

• Substitution: Replace variable with known value to find result

5 Parallel and Perpendicular Lines
Exercise 5
Find the equation of a line that is parallel to y = 3x - 4 and passes through the point (2, 1). Then find a line perpendicular to the original line passing through the same point.
Definition:

Parallel Lines: Lines with the same slope but different y-intercepts. Perpendicular Lines: Lines whose slopes are negative reciprocals of each other (m₁ × m₂ = -1).

Original Slope
m = 3
Parallel Line
y = 3x - 5
Perpendicular Line
y = (-1/3)x + 5/3
Step 1: Identify the slope of the original line

For y = 3x - 4, the slope is m = 3

Step 2: Find parallel line equation

Parallel lines have the same slope: m_parallel = 3

Using point (2, 1): y - 1 = 3(x - 2)

y - 1 = 3x - 6

y = 3x - 5

Step 3: Find perpendicular line equation

Perpendicular slope: m_perp = -1/m = -1/3

Using point (2, 1): y - 1 = (-1/3)(x - 2)

y - 1 = (-1/3)x + 2/3

y = (-1/3)x + 2/3 + 1

y = (-1/3)x + 5/3

Parallel: y = 3x - 5, Perpendicular: y = (-1/3)x + 5/3
Final answer:

Parallel line: y = 3x - 5. Perpendicular line: y = (-1/3)x + 5/3.

Applied rules:

• Parallel Lines: Same slope, different y-intercept

• Perpendicular Lines: Slopes are negative reciprocals

• Point-Slope Form: y - y₁ = m(x - x₁)

Linear Equations Laws, Methods, and Properties
\(y = mx + b\)
Slope-Intercept Form
Key definitions:

Linear Function: A function whose graph is a straight line. It has the form f(x) = mx + b where m and b are constants.

Slope: The ratio of the vertical change to the horizontal change between any two points on a line. It represents the rate of change.

Y-intercept: The y-coordinate of the point where the line crosses the y-axis.

Complete methodology:
  1. Identify the form needed: Choose based on given information
  2. Extract information: Determine slope, intercepts, or points
  3. Apply the appropriate formula: Use the correct form
  4. Solve for unknowns: Find missing values
  5. Convert if necessary: Change between forms as needed
  6. Verify the solution: Check against given conditions
Tip 1: Slope tells you the direction and steepness of the line.
Tip 2: Horizontal lines have slope 0, vertical lines have undefined slope.

Tip 3: Always check your equation by substituting known points.

Common errors: Confusing slope and y-intercept, incorrect sign when calculating slope, arithmetic mistakes in conversion, misidentifying variables in word problems.
Exam preparation: Practice converting between forms, master slope calculation, understand parallel/perpendicular relationships, work with real-world applications.

Linear equation properties:

• Slope-Intercept: y = mx + b

• Point-Slope: y - y₁ = m(x - x₁)

• Standard: Ax + By = C

• Parallel Lines: Same slope (m₁ = m₂)

• Perpendicular Lines: Negative reciprocal slopes (m₁ × m₂ = -1)

• Horizontal Line: y = b (slope = 0)

• Vertical Line: x = a (slope undefined)

Exercise with Visualization: Linear Relationships
Exercise 6: Different Slopes
Consider lines with different slopes passing through the origin:
y = x (slope 1), y = 2x (slope 2), y = 0.5x (slope 0.5), y = -x (slope -1)

Analysis: The chart shows how different slopes affect line steepness and direction.

  • Greater absolute value of slope = steeper line
  • Positive slope = upward trend
  • Negative slope = downward trend

Questions & Answers

Question: How do I know which form of linear equation to use in different situations?

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

  • Slope-Intercept (y = mx + b): When you know the slope and y-intercept
  • Point-Slope (y - y₁ = m(x - x₁)): When you know one point and the slope
  • Standard (Ax + By = C): When you want integer coefficients or need to find intercepts easily

Example:

  • Slope = 2, y-int = 3 → y = 2x + 3 (slope-intercept)
  • Point (1, 4), slope = 3 → y - 4 = 3(x - 1) (point-slope)
  • Need intercepts → Convert to Ax + By = C (standard)

You can always convert between forms as needed!

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

Answer: The slope represents the rate of change in real-life situations:

  • Distance-Time: Slope = speed (distance per unit time)
  • Cost-Quantity: Slope = price per item (cost per unit)
  • Temperature-Time: Slope = rate of temperature change
  • Earnings-Hours: Slope = hourly wage (earnings per hour)

In the taxi example from Exercise 4, the slope of 2 meant $2 per mile - the cost increases by $2 for each additional mile traveled.

Slope is fundamentally about how much one quantity changes when another quantity changes!

Question: Why do perpendicular lines have slopes that are negative reciprocals?

Answer: This relationship comes from geometry and rotation. When you rotate a line by 90°, the rise and run swap places and one changes sign.

If the original slope is m = rise/run, then the perpendicular slope becomes m_perp = -run/rise = -1/m.

For example:

  • Line with slope 2 → Perpendicular slope = -1/2
  • Line with slope -3 → Perpendicular slope = 1/3
  • Line with slope 1/4 → Perpendicular slope = -4

You can verify: m × m_perp = m × (-1/m) = -1

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

Answer: Follow these steps:

  1. Calculate the slope using m = (y₂ - y₁)/(x₂ - x₁)
  2. Use one of the points and the slope in point-slope form: y - y₁ = m(x - x₁)
  3. Convert to slope-intercept form if needed: y = mx + b

Example: Points (2, 3) and (4, 7)

  • Slope: m = (7 - 3)/(4 - 2) = 4/2 = 2
  • Using (2, 3): y - 3 = 2(x - 2)
  • Simplify: y - 3 = 2x - 4 → y = 2x - 1

The order of points doesn't matter - you'll get the same slope and equation!

Question: What's the difference between linear equations and linear functions?

Answer: The difference is:

  • Linear Equation: An equation like 2x + 3y = 6 that represents a relationship between variables
  • Linear Function: A function like f(x) = 2x + 3 where each input has exactly one output

All linear functions are linear equations, but not all linear equations are functions. The equation x = 3 is linear but not a function (fails vertical line test).

The function form y = mx + b is a specific type of linear equation that passes the vertical line test.