Solved Exercises on Graphing Linear Equations in Grade 7

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

Solution: Exercises 1 to 3
1 Slope-Intercept Form
Exercise 1
Graph the equation y = 2x + 3 using the slope-intercept method.
Definition:

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

Slope-intercept method:
  1. Identify the y-intercept (b) and plot the point (0, b)
  2. Identify the slope (m) and use it to find another point
  3. Draw a line through the two points
Equation
y = 2x + 3
Y-intercept
b = 3 → (0, 3)
Slope
m = 2 → rise = 2, run = 1
Step 1: Identify y-intercept

In y = 2x + 3, b = 3

Plot the y-intercept: (0, 3)

Step 2: Identify slope

In y = 2x + 3, m = 2

This means rise = 2 and run = 1

Step 3: Find second point

Starting from (0, 3), move up 2 and right 1

New point: (0 + 1, 3 + 2) = (1, 5)

Step 4: Draw the line

Draw a straight line through (0, 3) and (1, 5)

Step 5: Verify with third point

Check point (2, 7): y = 2(2) + 3 = 7 ✓

Line through (0, 3) and (1, 5)
Final answer:

Graph the line with y-intercept at (0, 3) and slope of 2.

Applied rules:

• Slope-intercept form: y = mx + b

• Y-intercept: Point where x = 0

• Slope interpretation: Rise over run

2 Using T-Table Method
Exercise 2
Graph the equation y = -x + 4 by creating a t-table and plotting points.
Definition:

T-table method: Creating a table of x and y values to plot ordered pairs and draw the line.

Equation
y = -x + 4
T-Table
x: -1, 0, 1, 2; y: 5, 4, 3, 2
Plotted Points
(-1, 5), (0, 4), (1, 3), (2, 2)
Step 1: Create the t-table

Choose convenient x-values (typically including 0)

Calculate corresponding y-values using the equation

Step 2: Calculate y-values

When x = -1: y = -(-1) + 4 = 5 → (-1, 5)

When x = 0: y = -(0) + 4 = 4 → (0, 4)

When x = 1: y = -(1) + 4 = 3 → (1, 3)

When x = 2: y = -(2) + 4 = 2 → (2, 2)

Step 3: Plot the points

Plot (-1, 5), (0, 4), (1, 3), and (2, 2)

Step 4: Draw the line

Connect the points with a straight line

Step 5: Verify the line

Check that the line passes through all plotted points

Line through (-1, 5), (0, 4), (1, 3), (2, 2)
Final answer:

Graph the line using the points from the t-table.

Applied rules:

• T-table method: Choose x-values, calculate y-values

• Ordered pairs: Plot (x, y) coordinates

• Linear verification: Three points should lie on the same straight line

3 Finding Intercepts Method
Exercise 3
Graph the equation 2x + 3y = 12 by finding the x-intercept and y-intercept.
Definition:

Intercept method: Find where the line crosses each axis, then draw the line through these points.

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

Draw a straight line through both intercepts

Step 5: Verify with third point

Check point (3, 2): 2(3) + 3(2) = 6 + 6 = 12 ✓

Line through (0, 4) and (6, 0)
Final answer:

Graph the line by connecting the x-intercept (6, 0) and y-intercept (0, 4).

Applied rules:

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

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

• Two-point method: Two points determine a unique line

Graphing Linear Equations Methods
y = mx + b
Slope-Intercept Form
Slope-Intercept
y = mx + b
Use y-intercept and slope
T-Table
Choose x, calculate y
Plot ordered pairs
Intercept Method
Find x and y intercepts
Connect intercepts
Key definitions:

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

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

Y-intercept: 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).

T-table: A table showing corresponding x and y values for an equation.

Ordered pair: A pair of numbers (x, y) that represents a point on the coordinate plane.

Graphing methodologies:
  1. Slope-intercept method: Identify y-intercept and slope, plot and use slope to find additional points
  2. T-table method: Choose x-values, calculate y-values, plot points and draw line
  3. Intercept method: Find x and y intercepts, plot and connect them
  4. Standard form method: Convert to slope-intercept or use intercept method
Tip 1: Always plot at least 3 points to verify accuracy of the line.
Tip 2: For slope-intercept form, start with the y-intercept as it's easiest to locate.
Tip 3: When slope is a fraction, use rise over run to find the next point.
Tip 4: Verify your graph by checking that points satisfy the original equation.
Common errors: Mixing up x and y coordinates, incorrect slope calculation, not plotting points accurately, confusing positive and negative slopes, forgetting to extend the line in both directions.
Exam preparation: Practice all three methods, master coordinate plotting, understand slope direction, work with different forms of equations, practice graphing on grid paper.
Essential rules:

• Slope-intercept form: y = mx + b

• Y-intercept: value of y when x = 0

• X-intercept: value of x when y = 0

• Positive slope: line rises from left to right

• Negative slope: line falls from left to right

• Two points determine a unique line

Solution: Exercises 4 to 5
4 Standard Form Graphing
Exercise 4
Graph the equation 3x - 2y = 6 by converting to slope-intercept form.
Definition:

Standard form: Ax + By = C, where A, B, and C are integers. Can be converted to slope-intercept form.

Original Equation
3x - 2y = 6
Convert to Slope-Intercept
y = (3/2)x - 3
Identify Components
Slope = 3/2, Y-int = -3
Step 1: Convert to slope-intercept form

Start with: 3x - 2y = 6

Subtract 3x from both sides: -2y = -3x + 6

Divide by -2: y = (3/2)x - 3

Step 2: Identify y-intercept

In y = (3/2)x - 3, b = -3

Y-intercept: (0, -3)

Step 3: Identify slope

In y = (3/2)x - 3, m = 3/2

This means rise = 3 and run = 2

Step 4: Find second point

Starting from (0, -3), move up 3 and right 2

New point: (0 + 2, -3 + 3) = (2, 0)

Step 5: Draw the line

Draw a straight line through (0, -3) and (2, 0)

Line through (0, -3) and (2, 0)
Final answer:

Graph the line with equation y = (3/2)x - 3, which passes through (0, -3) and (2, 0).

Applied rules:

• Standard to slope-intercept: Solve for y to get y = mx + b

• Fractional slope: Rise over run (numerator over denominator)

• Conversion method: Convert difficult forms to easier ones

5 Real-World Application
Exercise 5
A company's profit P (in thousands of dollars) is related to the number of items sold x by P = 1.5x - 10. Graph this relationship and interpret the intercepts.
Definition:

Real-world applications: Linear equations model many practical situations with constant rates of change.

Equation
P = 1.5x - 10
Y-intercept
(0, -10)
X-intercept
(6.67, 0)
Step 1: Identify the equation form

P = 1.5x - 10 is in slope-intercept form

Slope = 1.5, Y-intercept = -10

Step 2: Find y-intercept

When x = 0: P = 1.5(0) - 10 = -10

Y-intercept: (0, -10) - Loss of $10,000 when no items sold

Step 3: Find x-intercept

When P = 0: 0 = 1.5x - 10

1.5x = 10

x = 6.67

X-intercept: (6.67, 0) - Break-even at about 7 items

Step 4: Plot the points

Plot (0, -10) and (6.67, 0), then draw the line

Step 5: Interpret the graph

Slope = 1.5: Profit increases by $1,500 per item sold

Y-intercept: Fixed costs of $10,000

X-intercept: Break-even point

Line through (0, -10) and (6.67, 0)
Final answer:

Graph shows the profit function with y-intercept (0, -10) representing fixed costs and x-intercept (6.67, 0) representing the break-even point.

Applied rules:

• Real-world interpretation: Intercepts have contextual meaning

• Slope interpretation: Represents rate of change in context

• Modeling: Linear equations represent real-world relationships

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

Linear equation: An equation that forms a straight line when graphed. The variables have an exponent of 1.

Slope-intercept form: y = mx + b, where m represents the slope (steepness) and b represents the y-intercept (starting point).

Standard form: Ax + By = C, where A, B, and C are integers, and A is typically positive.

Point-slope form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.

Coordinate plane: A two-dimensional plane with x and y axes used for graphing points and equations.

Ordered pair: A pair of numbers (x, y) that represents a location on the coordinate plane.

Complete graphing methodology:
  1. Identify the form: Determine if equation is in slope-intercept, standard, or point-slope form
  2. Choose method: Select the most efficient graphing method based on the equation form
  3. Find points: Calculate or identify at least two points on the line
  4. Plot points: Accurately mark the points on the coordinate plane
  5. Draw line: Connect points with a straight edge and extend in both directions
  6. Verify: Check that additional points satisfy the equation
Tip 1: When slope is negative, the line falls from left to right; when positive, it rises.
Tip 2: Always extend your line beyond the plotted points to show the complete linear relationship.
Tip 3: Use intercepts when available as they're often easy to calculate and plot.
Tip 4: Check your work by substituting coordinates back into the original equation.
Common errors: Mixing up x and y coordinates, incorrect slope calculation, not extending the line fully, plotting points inaccurately, confusing positive and negative slopes, arithmetic errors when calculating points.
Applications: Cost calculations, distance-speed-time problems, conversion formulas, business profit analysis, scientific measurements, and modeling any situation with constant rate of change.
Essential graphing rules:

• Slope-intercept form: y = mx + b

• Standard form: Ax + By = C

• Y-intercept: (0, b) where line crosses y-axis

• X-intercept: (a, 0) where line crosses x-axis

• Positive slope: line rises left to right

• Negative slope: line falls left to right

• Horizontal line: y = constant (slope = 0)

• Vertical line: x = constant (slope undefined)

Graphing Linear Equations Guide

📊
Slope-Intercept

y = mx + b

Plot (0, b)

Use slope m to find next point

T-Table Method

Choose x values

Calculate y values

Plot (x, y) points

Intercept Method

Find x-intercept (y = 0)

Find y-intercept (x = 0)

Connect the two points

Verification

Check points satisfy equation

Extend line in both directions

Verify accuracy

Questions & Answers

Question: How do I know which graphing method to use?

Answer: Choose the method based on the equation form:

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

  • Use the y-intercept (0, b) and slope (m) method
  • Easiest when equation is already in this form
  • Quick to identify y-intercept and direction

Standard form (Ax + By = C):

  • Use the intercept method (find x and y intercepts)
  • Alternatively, convert to slope-intercept form
  • Intercepts are often whole numbers

Any form:

  • T-table method works for any linear equation
  • Choose when other methods seem complicated
  • Especially useful when slope is a fraction

Choose the method that makes the fewest calculations!

Question: What if the slope is a fraction like 3/4? How do I use it to find points?

Answer: When slope is a fraction, use the rise over run concept:

For slope = 3/4:

  • Rise = 3 (move up 3 units)
  • Run = 4 (move right 4 units)

Example: Starting from point (0, 2) with slope 3/4

  • From (0, 2), move up 3 and right 4
  • New point: (0 + 4, 2 + 3) = (4, 5)

Negative fractional slope:

  • For slope = -3/4, rise = -3 (move down 3) and run = 4 (move right 4)
  • Or rise = 3 (move up 3) and run = -4 (move left 4)

The numerator tells you how far to move vertically, the denominator tells you how far horizontally!

Question: Why do I need to extend the line beyond the points I plotted?

Answer: The line represents ALL possible solutions to the equation, not just the points you calculated!

Key reasons to extend the line:

  • All solutions: Every point on the line is a solution to the equation
  • Continuity: The relationship continues infinitely in both directions
  • Prediction: You can estimate values beyond your calculated points
  • Completeness: Shows the full linear relationship

Example: In y = 2x + 1, if you only drew the line between (0, 1) and (1, 3), you'd miss solutions like (5, 11) or (-2, -3).

Real-world context:

  • Cost equations continue beyond your sample points
  • Distance-time relationships continue beyond measured values
  • The linear pattern extends indefinitely

A complete line shows the full relationship between the variables!