Graphing Linear Equations

Graph lines using slope-intercept form

Graphing Linear Equations

What is a Graph of a Linear Equation?

The graph of a linear equation is a straight line on the coordinate plane. Every point (x, y) on the line is a solution to the equation.

For example, the equation y = 2x + 1 is graphed as a line where every point satisfies the equation. The point (2, 5) is on the line because 5 = 2(2) + 1.

The Coordinate Plane

The coordinate plane (also called the Cartesian plane) has:

A horizontal axis called the x-axis
A vertical axis called the y-axis
The point where they meet is the origin (0, 0)

The plane is divided into four quadrants:

Quadrant I: (+, +) upper right
Quadrant II: (-, +) upper left
Quadrant III: (-, -) lower left
Quadrant IV: (+, -) lower right

Ordered Pairs

Points are written as (x, y) called ordered pairs:

The first number is the x-coordinate (horizontal position)
The second number is the y-coordinate (vertical position)

To plot (3, 4):

Start at origin
Move 3 units right (positive x)
Move 4 units up (positive y)
Mark the point

To plot (-2, 5):

Start at origin
Move 2 units left (negative x)
Move 5 units up (positive y)
Mark the point

Graphing Linear Equations Using a Table

Method: Make a table of values, plot points, draw a line

Example: Graph y = 2x - 3

Step 1: Create a table (choose x-values, calculate y)

| x | y = 2x - 3 | (x, y) | |----|------------|-----------| | -1 | 2(-1) - 3 = -5 | (-1, -5) | | 0 | 2(0) - 3 = -3 | (0, -3) | | 1 | 2(1) - 3 = -1 | (1, -1) | | 2 | 2(2) - 3 = 1 | (2, 1) | | 3 | 2(3) - 3 = 3 | (3, 3) |

Step 2: Plot these points on a coordinate plane

Step 3: Draw a straight line through the points

Step 4: Add arrows at both ends (line extends forever)

Tip: You only need 2 points to draw a line, but use 3 as a check!

Slope-Intercept Form

The most common form for graphing is slope-intercept form:

y = mx + b

Where:

m is the slope (steepness of line)
b is the y-intercept (where line crosses y-axis)

Example: y = 3x + 2

Slope (m) = 3
Y-intercept (b) = 2

Understanding Slope

Slope measures the steepness and direction of a line.

Slope = rise/run = change in y / change in x = (y₂ - y₁)/(x₂ - x₁)

Types of Slope:

Positive Slope (m > 0):

Line goes upward from left to right
Example: y = 2x + 1 (slope = 2)

Negative Slope (m < 0):

Line goes downward from left to right
Example: y = -3x + 5 (slope = -3)

Zero Slope (m = 0):

Horizontal line
Example: y = 4 (slope = 0)

Undefined Slope:

Vertical line
Example: x = 3 (slope undefined)

Graphing Using Slope-Intercept Form

Method: Start at y-intercept, use slope to find next points

Example: Graph y = (2/3)x - 1

Step 1: Identify slope and y-intercept

Slope m = 2/3 (rise 2, run 3)
Y-intercept b = -1

Step 2: Plot y-intercept Plot point (0, -1)

Step 3: Use slope to find next point From (0, -1):

Rise 2 (up 2 units)
Run 3 (right 3 units)
Plot point (3, 1)

Step 4: Continue or go backwards From (0, -1):

Fall 2 (down 2)
Run left 3 (left 3)
Plot point (-3, -3)

Step 5: Draw line through points

Finding Slope from Two Points

Given two points (x₁, y₁) and (x₂, y₂):

slope = (y₂ - y₁)/(x₂ - x₁)

Example: Find slope through (1, 3) and (4, 9) m = (9 - 3)/(4 - 1) m = 6/3 m = 2

The Y-Intercept

The y-intercept is where the line crosses the y-axis.

At this point, x = 0
Written as point (0, b)

To find y-intercept from an equation: Set x = 0 and solve for y

Example: Find y-intercept of 2x + 3y = 12 Set x = 0: 2(0) + 3y = 12 3y = 12 y = 4 Y-intercept: (0, 4)

The X-Intercept

The x-intercept is where the line crosses the x-axis.

At this point, y = 0
Written as point (a, 0)

To find x-intercept from an equation: Set y = 0 and solve for x

Example: Find x-intercept of 2x + 3y = 12 Set y = 0: 2x + 3(0) = 12 2x = 12 x = 6 X-intercept: (6, 0)

Graphing Using Intercepts

Method: Find x-intercept and y-intercept, draw line through them

Example: Graph 3x + 2y = 12

Step 1: Find y-intercept (set x = 0) 3(0) + 2y = 12 y = 6 Point: (0, 6)

Step 2: Find x-intercept (set y = 0) 3x + 2(0) = 12 x = 4 Point: (4, 0)

Step 3: Plot both intercepts

Step 4: Draw line through them

Step 5: Check with a third point

Standard Form

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

Example: 2x + 3y = 12

To graph from standard form:

Find x and y intercepts, OR
Convert to slope-intercept form

Converting to slope-intercept form: 2x + 3y = 12 3y = -2x + 12 y = (-2/3)x + 4

Special Lines

Horizontal Lines:

Form: y = k (constant)
Slope = 0
Parallel to x-axis
Example: y = 3

Vertical Lines:

Form: x = k (constant)
Undefined slope
Parallel to y-axis
Example: x = -2

Parallel and Perpendicular Lines

Parallel Lines:

Have the SAME slope
Never intersect
Example: y = 2x + 1 and y = 2x - 3 (both have slope 2)

Perpendicular Lines:

Slopes are NEGATIVE RECIPROCALS
Intersect at 90° angle
If slope₁ = m, then slope₂ = -1/m
Example: y = 2x + 1 (slope = 2) and y = (-1/2)x + 3 (slope = -1/2)

Product of perpendicular slopes = -1

Common Mistakes to Avoid

Plotting points incorrectly Remember: (x, y) not (y, x)!
Wrong direction for slope Positive slope: up-right Negative slope: down-right
Not extending line far enough Lines go on forever - add arrows!
Confusing slope and y-intercept In y = 3x + 2: slope is 3, intercept is 2
Calculating slope incorrectly Use (y₂ - y₁)/(x₂ - x₁), keep order consistent

Real-World Applications

Example 1: Phone Plan A phone plan costs $20/month plus$ 0.10 per text. Equation: y = 0.10x + 20

Slope: $0.10 per text
Y-intercept: $20 base fee

Example 2: Temperature Converting Fahrenheit to Celsius: C = (5/9)(F - 32) This is a linear relationship.

Example 3: Car Depreciation A car worth $30,000 depreciates$ 2,000/year. Equation: y = -2000x + 30000

Slope: -$2,000/year (negative = decreasing)
Y-intercept: $30,000 initial value

Problem-Solving Strategy

When given a word problem:

Identify rate of change (slope)
Identify starting value (y-intercept)
Write equation in y = mx + b form
Graph the equation
Use graph to answer questions

Quick Reference

| Form | Equation | Use | |------|----------|-----| | Slope-Intercept | y = mx + b | Easy to graph, see slope/intercept | | Standard | Ax + By = C | Find intercepts easily | | Point-Slope | y - y₁ = m(x - x₁) | Know slope and one point | | Horizontal | y = k | Slope is 0 | | Vertical | x = k | Undefined slope |

Practice Tips

Always label axes and scale
Plot at least 3 points to verify
Use a ruler for straight lines
Check your points in the equation
Extend line with arrows
Practice recognizing slope from graphs

📚 Practice Problems

1Problem 1easy

❓ Question:

What is the slope and y-intercept of y = 4x - 3?

💡 Show Solution

Step 1: Recognize the slope-intercept form: y = mx + b where m is the slope and b is the y-intercept

Step 2: Identify m and b in y = 4x - 3: Comparing to y = mx + b: m = 4 (the coefficient of x) b = -3 (the constant term)

Step 3: State the answers: Slope = 4 (or 4/1, meaning rise 4, run 1) y-intercept = -3 (the point (0, -3))

Answer: Slope = 4, y-intercept = -3

2Problem 2easy

❓ Question:

Find the slope and y-intercept of the line: y = 3x + 2

💡 Show Solution

This is in slope-intercept form: y = mx + b

Comparing y = 3x + 2 to y = mx + b: m = 3 (slope) b = 2 (y-intercept)

Slope: 3 (or 3/1, meaning rise 3, run 1) Y-intercept: 2 (point is (0, 2))

3Problem 3easy

❓ Question:

Graph the equation y = -2x + 4 using the slope and y-intercept.

💡 Show Solution

Step 1: Identify slope and y-intercept y = -2x + 4 Slope (m) = -2 = -2/1 (down 2, right 1) Y-intercept (b) = 4

Step 2: Plot y-intercept at (0, 4)

Step 3: Use slope to find another point From (0, 4), go down 2 and right 1 to get (1, 2)

Step 4: Draw line through (0, 4) and (1, 2)

The line slopes downward from left to right.

4Problem 4easy

❓ Question:

Graph the equation y = -2x + 1

💡 Show Solution

Step 1: Identify slope and y-intercept: Slope m = -2 (or -2/1) y-intercept b = 1 (point (0, 1))

Step 2: Plot the y-intercept: Start at (0, 1) on the graph

Step 3: Use the slope to find another point: Slope = -2/1 means: rise -2, run 1 From (0, 1): go down 2, right 1 → (1, -1)

Step 4: Plot the second point at (1, -1)

Step 5: Draw a line through both points

Points on the line: (0, 1), (1, -1), (2, -3), etc.

Answer: A line passing through (0, 1) with slope -2

5Problem 5easy

❓ Question:

Find the slope and y-intercept of the line $y = -3x + 5$

💡 Show Solution

The equation is already in slope-intercept form $y = mx + b$

Compare $y = -3x + 5$ with $y = mx + b$ :

Slope: $m = -3$
Y-intercept: $b = 5$

This means:

The line has a slope of $-3$ (goes down 3 units for every 1 unit to the right)
The line crosses the y-axis at the point $(0, 5)$

Answer: Slope = $-3$ , y-intercept = $5$

6Problem 6medium

❓ Question:

Find the slope of the line passing through points (2, 3) and (6, 11).

💡 Show Solution

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

Point 1: (2, 3) → x₁ = 2, y₁ = 3 Point 2: (6, 11) → x₂ = 6, y₂ = 11

m = (11 - 3)/(6 - 2) m = 8/4 m = 2

The slope is 2.

7Problem 7medium

❓ Question:

Find the slope of the line passing through (2, 5) and (6, 13)

💡 Show Solution

Step 1: Use the slope formula: m = (y₂ - y₁)/(x₂ - x₁)

Step 2: Identify the coordinates: Point 1: (x₁, y₁) = (2, 5) Point 2: (x₂, y₂) = (6, 13)

Step 3: Substitute into the formula: m = (13 - 5)/(6 - 2) m = 8/4 m = 2

Step 4: Interpret the slope: For every 1 unit right, the line goes up 2 units

Answer: The slope is 2

8Problem 8medium

❓ Question:

Find the slope of the line passing through $(2, 5)$ and $(6, 13)$

💡 Show Solution

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

Identify the points: $(x_1, y_1) = (2, 5)$ and $(x_2, y_2) = (6, 13)$

Substitute: $m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$

Answer: The slope is $2$

9Problem 9medium

❓ Question:

Write the equation of a line with slope 4 that passes through the point (1, 5).

💡 Show Solution

Use point-slope form: y - y₁ = m(x - x₁)

Given: m = 4, point (1, 5)

y - 5 = 4(x - 1)

Distribute: y - 5 = 4x - 4

Add 5 to both sides: y = 4x + 1

Answer: y = 4x + 1

10Problem 10medium

❓ Question:

Convert $4x - 2y = 8$ to slope-intercept form

💡 Show Solution

We need to solve for $y$ to get the form $y = mx + b$

Step 1: Subtract $4x$ from both sides $-2y = -4x + 8$

Step 2: Divide everything by $-2$ $y = \frac{-4x + 8}{-2}$ $y = 2x - 4$

Answer: $y = 2x - 4$ (slope = $2$ , y-intercept = $-4$ )

11Problem 11medium

❓ Question:

Write the equation of a line with slope -3 that passes through (2, 1)

💡 Show Solution

Step 1: Use point-slope form: y - y₁ = m(x - x₁)

Step 2: Substitute m = -3 and point (2, 1): y - 1 = -3(x - 2)

Step 3: Distribute the -3: y - 1 = -3x + 6

Step 4: Solve for y (slope-intercept form): y = -3x + 6 + 1 y = -3x + 7

Step 5: Check: Does (2, 1) satisfy the equation? 1 = -3(2) + 7 1 = -6 + 7 1 = 1 ✓

Answer: y = -3x + 7

12Problem 12hard

❓ Question:

Write the equation of the line passing through (-1, 4) and (3, -2)

💡 Show Solution

Step 1: Find the slope: m = (y₂ - y₁)/(x₂ - x₁) m = (-2 - 4)/(3 - (-1)) m = -6/4 m = -3/2

Step 2: Use point-slope form with either point (using (-1, 4)): y - 4 = (-3/2)(x - (-1)) y - 4 = (-3/2)(x + 1)

Step 3: Distribute: y - 4 = (-3/2)x - 3/2

Step 4: Solve for y: y = (-3/2)x - 3/2 + 4 y = (-3/2)x - 3/2 + 8/2 y = (-3/2)x + 5/2

Step 5: Check with both points: Point (-1, 4): 4 = (-3/2)(-1) + 5/2 = 3/2 + 5/2 = 8/2 = 4 ✓ Point (3, -2): -2 = (-3/2)(3) + 5/2 = -9/2 + 5/2 = -4/2 = -2 ✓

Answer: y = (-3/2)x + 5/2 or y = -1.5x + 2.5

13Problem 13hard

❓ Question:

Find the equation of the line passing through (3, 7) and (5, 13) in slope-intercept form.

💡 Show Solution

Step 1: Find the slope m = (13 - 7)/(5 - 3) = 6/2 = 3

Step 2: Use point-slope form with either point (using (3, 7)): y - 7 = 3(x - 3)

Step 3: Convert to slope-intercept form y - 7 = 3x - 9 y = 3x - 2

Answer: y = 3x - 2

Check with other point (5, 13): y = 3(5) - 2 = 15 - 2 = 13 ✓

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