Solved Exercises on Writing Equations from Graphs in Grade 7

Master writing equations from graphs: slope-intercept form, standard form, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 From Y-Intercept and Slope
Exercise 1
Write the equation of the line shown in the graph where the y-intercept is (0, 3) and the slope is 2.
Definition:

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

Writing equation method:
  1. Identify the y-intercept (b) from the graph
  2. Calculate the slope (m) from the graph
  3. Substitute m and b into y = mx + b
Given Information
y-intercept = 3, slope = 2
Substitute into form
y = 2x + 3
Final Equation
y = 2x + 3
Step 1: Identify y-intercept

The line crosses the y-axis at (0, 3)

Therefore, b = 3

Step 2: Identify slope

From the graph, the slope is 2 (rise of 2 for every run of 1)

Therefore, m = 2

Step 3: Write the equation

Substitute m = 2 and b = 3 into y = mx + b

y = 2x + 3

Step 4: Verify the equation

Check point (1, 5): y = 2(1) + 3 = 5 ✓

Check y-intercept (0, 3): y = 2(0) + 3 = 3 ✓

y = 2x + 3
Final answer:

The equation of the line is y = 2x + 3.

Applied rules:

• Slope-intercept form: y = mx + b

• Y-intercept: Point where x = 0

• Slope: Rise over run

2 From Two Points
Exercise 2
Write the equation of the line passing through points (1, 4) and (3, 8) shown in the graph.
Definition:

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

Given Points
(1, 4) and (3, 8)
Calculate Slope
m = (8-4)/(3-1) = 2
Write Equation
y = 2x + 2
Step 1: Calculate the slope

Use the slope formula: m = (y₂ - y₁)/(x₂ - x₁)

m = (8 - 4)/(3 - 1) = 4/2 = 2

Step 2: Use point-slope form

Using point (1, 4) and slope m = 2

y - 4 = 2(x - 1)

Step 3: Convert to slope-intercept form

y - 4 = 2x - 2

y = 2x - 2 + 4

y = 2x + 2

Step 4: Verify the equation

Check point (1, 4): y = 2(1) + 2 = 4 ✓

Check point (3, 8): y = 2(3) + 2 = 8 ✓

y = 2x + 2
Final answer:

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

Applied rules:

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

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

• Conversion: Point-slope to slope-intercept form

3 From Intercepts
Exercise 3
Write the equation of the line with x-intercept (4, 0) and y-intercept (0, 6) shown in the graph.
Definition:

Intercept form: When given both intercepts, use them to find slope and write the equation.

Given Intercepts
X-int: (4, 0), Y-int: (0, 6)
Calculate Slope
m = (6-0)/(0-4) = -3/2
Write Equation
y = -3/2x + 6
Step 1: Identify the intercepts

X-intercept: (4, 0) - where line crosses x-axis

Y-intercept: (0, 6) - where line crosses y-axis

Step 2: Calculate the slope

Use the slope formula with the two intercepts:

m = (6 - 0)/(0 - 4) = 6/(-4) = -3/2

Step 3: Write in slope-intercept form

Since y-intercept is (0, 6), b = 6

Since slope m = -3/2

y = (-3/2)x + 6

y = -3/2x + 6

Step 4: Verify the equation

Check x-intercept (4, 0): y = -3/2(4) + 6 = -6 + 6 = 0 ✓

Check y-intercept (0, 6): y = -3/2(0) + 6 = 6 ✓

y = -3/2x + 6
Final answer:

The equation of the line is y = -3/2x + 6.

Applied rules:

• Y-intercept: Directly gives b value in y = mx + b

• Slope from intercepts: Use both intercepts as two points

• Negative slope: Line falls from left to right

Writing Equations from Graphs Methods
y = mx + b
Slope-Intercept Form
Slope-Intercept
y = mx + b
Identify slope and y-intercept
Point-Slope
y - y₁ = m(x - x₁)
Use point and slope
Standard Form
Ax + By = C
Convert from other forms
Key definitions:

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

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

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).

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

Graph interpretation: Reading information from a visual representation of a linear relationship.

Writing equation methodology:
  1. Observe the graph: Identify key features like intercepts and direction
  2. Find the slope: Calculate rise over run or identify from the graph
  3. Find the y-intercept: Locate where the line crosses the y-axis
  4. Substitute values: Put slope and y-intercept into y = mx + b
  5. Verify: Check that the equation fits the graph
Tip 1: Always start by identifying the y-intercept as it's often easiest to spot on the graph.
Tip 2: For slope, count boxes on the graph to determine rise and run.
Tip 3: Positive slope means the line rises from left to right; negative slope means it falls.
Tip 4: Always verify your equation by testing points from the graph.
Common errors: Reading coordinates incorrectly, miscalculating slope, confusing positive and negative slopes, not identifying the correct y-intercept, arithmetic errors when converting forms.
Exam preparation: Practice identifying key features from graphs, master slope calculation, understand the relationship between graph direction and slope sign, work with fractional slopes.
Essential rules:

• Slope-intercept form: y = mx + b

• Y-intercept: value of y when x = 0

• Slope: rise over run (change in y divided by change in x)

• Positive slope: line rises left to right

• Negative slope: line falls left to right

• Zero slope: horizontal line

Solution: Exercises 4 to 5
4 Converting to Standard Form
Exercise 4
Write the equation of the line shown in slope-intercept form, then convert it to standard form.
Definition:

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

Slope-Intercept
y = 3x - 4
Convert to Standard
3x - y = 4
Final Form
3x - y = 4
Step 1: Write in slope-intercept form

From the graph, y-intercept is -4 and slope is 3

y = 3x - 4

Step 2: Convert to standard form

Start with: y = 3x - 4

Subtract 3x from both sides: -3x + y = -4

Multiply by -1 to make A positive: 3x - y = 4

Step 3: Verify standard form

A = 3, B = -1, C = 4 (all integers)

A is positive ✓

Step 4: Check with a point

Using point (2, 2): 3(2) - 2 = 6 - 2 = 4 ✓

3x - y = 4
Final answer:

The equation in standard form is 3x - y = 4.

Applied rules:

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

• Positive A: Standard form typically has A > 0

• Algebraic manipulation: Rearrange terms to achieve standard form

5 Real-World Application
Exercise 5
A graph shows the relationship between the number of hours worked (x) and total earnings (y) for a job. The line passes through (0, 50) and (4, 170). Write the equation and interpret the y-intercept.
Definition:

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

Given Points
(0, 50) and (4, 170)
Calculate Slope
m = (170-50)/(4-0) = 30
Write Equation
y = 30x + 50
Step 1: Calculate the slope

m = (170 - 50)/(4 - 0) = 120/4 = 30

This means $30 per hour

Step 2: Identify y-intercept

Point (0, 50) shows that when x = 0, y = 50

Therefore, b = 50

Step 3: Write the equation

Substituting m = 30 and b = 50:

y = 30x + 50

Step 4: Interpret the y-intercept

When x = 0 (no hours worked), y = 50

This represents a base payment or signing bonus of $50

Step 5: Verify the equation

Check (4, 170): y = 30(4) + 50 = 120 + 50 = 170 ✓

y = 30x + 50; y-intercept = $50 base payment
Final answer:

The equation is y = 30x + 50, where the y-intercept of 50 represents a $50 base payment before any hours are worked.

Applied rules:

• Real-world interpretation: Y-intercept often represents initial value

• Slope interpretation: Rate of change in context

• Modeling: Linear equations represent real-world relationships

Detailed Writing Equations from Graphs Guide
y = mx + b
Slope-Intercept Form
Key definitions:

Writing equations from graphs: The process of analyzing a visual representation of a linear relationship to determine its algebraic form.

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

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

Linear function: A function that produces a straight line when graphed, with a constant rate of change.

Graph interpretation: Reading quantitative information from a visual representation to extract mathematical relationships.

Coordinate identification: Locating and reading ordered pairs (x, y) from a graph accurately.

Complete writing equation methodology:
  1. Survey the graph: Observe the general shape, direction, and key points
  2. Identify y-intercept: Locate where the line crosses the y-axis
  3. Calculate slope: Determine rise over run using any two convenient points
  4. Substitute values: Plug slope and y-intercept into y = mx + b
  5. Verify accuracy: Test the equation with additional points from the graph
  6. Convert if needed: Change to required form (standard, point-slope, etc.)
Tip 1: Always start with the y-intercept as it's usually easiest to identify on the graph.
Tip 2: Use grid points for accurate slope calculation - avoid estimating between grid lines.
Tip 3: If the line falls from left to right, the slope is negative.
Tip 4: Always verify your equation by testing it with points from the original graph.
Common errors: Misreading coordinates, incorrect slope calculation (especially with negative slopes), confusing x and y intercepts, arithmetic errors when converting forms, not verifying the final equation.
Applications: Cost analysis, distance-speed-time problems, scientific measurements, business profit analysis, temperature conversions, and modeling any situation with constant rate of change.
Essential writing equation rules:

• Slope-intercept form: y = mx + b

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

• Y-intercept: value of y when x = 0

• Positive slope: line rises left to right

• Negative slope: line falls left to right

• Standard form: Ax + By = C (A, B, C are integers)

Writing Equations from Graphs Guide

📊
Identify Y-Intercept

Find where line crosses y-axis

This is the b value

Point: (0, b)

Calculate Slope

m = rise/run

Choose two clear points

m = (y₂ - y₁)/(x₂ - x₁)

Write Equation

y = mx + b

Substitute slope and y-intercept

Verify with graph points

Verification

Test multiple points

Check direction of line

Confirm slope and intercept

Questions & Answers

Question: How do I calculate slope when the points aren't at exact grid intersections?

Answer: When points aren't at exact grid intersections, you can still calculate slope using the formula:

Method 1 - Estimate coordinates:

  • Read the approximate coordinates from the graph
  • Use the slope formula: m = (y₂ - y₁)/(x₂ - x₁)
  • For example: if points are roughly at (1.5, 2.3) and (3.7, 6.1)
  • m = (6.1 - 2.3)/(3.7 - 1.5) = 3.8/2.2 ≈ 1.73

Method 2 - Count boxes:

  • Count the number of boxes (or fractions of boxes) vertically and horizontally
  • This gives you rise and run directly
  • For example: if you go up 2.5 boxes and right 1.5 boxes, m = 2.5/1.5 = 5/3

The key is to be as accurate as possible with your estimates!

Question: What if I can't see the y-intercept on the graph? How do I write the equation?

Answer: Even if the y-intercept isn't visible, you can still write the equation using any two points from the graph:

Step-by-step process:

  1. Identify two points: Find any two clear points on the line
  2. Calculate slope: Use m = (y₂ - y₁)/(x₂ - x₁)
  3. Use point-slope form: y - y₁ = m(x - x₁) with one of your points
  4. Solve for y: Rearrange to get slope-intercept form

Example: If you see points (2, 3) and (4, 7) on the line

  • Slope: m = (7 - 3)/(4 - 2) = 4/2 = 2
  • Point-slope: y - 3 = 2(x - 2)
  • Simplify: y - 3 = 2x - 4
  • Slope-intercept: y = 2x - 1

Now you can identify that the y-intercept is -1!

Question: How do I know if my equation is correct after writing it from the graph?

Answer: There are several ways to verify your equation:

Method 1 - Test points:

  • Pick several points that are clearly marked on the original graph
  • Substitute the x-coordinate into your equation
  • Check if the calculated y-value matches the y-coordinate on the graph

Method 2 - Check intercepts:

  • If you can see the y-intercept on the graph, substitute x = 0 into your equation
  • If you can see the x-intercept, substitute y = 0 and solve for x
  • Compare with the intercepts on the graph

Method 3 - Check slope:

  • Visually confirm that the direction of your line matches the sign of your slope
  • Check that the steepness of your line seems consistent with your calculated slope

Example: If your equation is y = 2x + 1 and the graph shows points (0, 1) and (1, 3)

  • Test (0, 1): y = 2(0) + 1 = 1 ✓
  • Test (1, 3): y = 2(1) + 1 = 3 ✓
  • Both points match, so the equation is correct!