Linear Equations - Grade 8

1. Is (x, y) a Solution to the Linear Equation?

Definition: A point (x, y) is a solution to a linear equation if, when you substitute the x and y values into the equation, it makes the equation true.

Steps to Check if (x, y) is a Solution:

Substitute the x-value into the equation
Substitute the y-value into the equation
Simplify both sides of the equation
If both sides are equal, it IS a solution; if not, it is NOT a solution

Examples:

Example 1: Is (2, 5) a solution to \( y = 3x - 1 \)?

Substitute x = 2 and y = 5:

\( 5 = 3(2) - 1 \)

\( 5 = 6 - 1 \)

\( 5 = 5 \) ✓

Yes, (2, 5) IS a solution.

Example 2: Is (1, 4) a solution to \( 2x + y = 7 \)?

Substitute x = 1 and y = 4:

\( 2(1) + 4 = 7 \)

\( 2 + 4 = 7 \)

\( 6 = 7 \) ✗

No, (1, 4) is NOT a solution.

2. Relate the Graph of an Equation to Its Solutions

Key Concept: Every point on the graph of a linear equation represents a solution to that equation. Conversely, every solution to the equation is a point on its graph.

Important Facts:

If a point is ON the line, it IS a solution
If a point is NOT on the line, it is NOT a solution
A linear equation has infinitely many solutions (all points on the line)
The graph of a linear equation is always a straight line

Finding Solutions from a Graph:

Locate points where the line crosses grid intersections
Read the x-coordinate and y-coordinate of each point
Write as an ordered pair (x, y)
All such points are solutions to the equation

Example:

If the line of \( y = 2x + 1 \) passes through points (0, 1), (1, 3), and (2, 5):

All three points are solutions
Any point NOT on the line (like (1, 1)) is NOT a solution

3. Slope-Intercept Form: Find the Slope and Y-Intercept

Slope-Intercept Form:

\( y = mx + b \)

where \( m \) = slope and \( b \) = y-intercept

What They Mean:

Slope (m): The rate of change; steepness of the line; coefficient of x
Y-intercept (b): Where the line crosses the y-axis; the constant term; point (0, b)

How to Identify Slope and Y-Intercept:

Make sure the equation is in \( y = mx + b \) form (y isolated on left side)
The coefficient of x is the slope (m)
The constant term is the y-intercept (b)

Examples:

Example 1: Find the slope and y-intercept of \( y = 4x + 7 \)

Slope: \( m = 4 \)

Y-intercept: \( b = 7 \) → Point (0, 7)

Example 2: Find the slope and y-intercept of \( y = -\frac{2}{3}x - 5 \)

Slope: \( m = -\frac{2}{3} \)

Y-intercept: \( b = -5 \) → Point (0, -5)

Example 3: Find the slope and y-intercept of \( y = x \)

Slope: \( m = 1 \) (coefficient of x is 1)

Y-intercept: \( b = 0 \) → Point (0, 0)

4. Graph a Line from an Equation in Slope-Intercept Form

Steps to Graph \( y = mx + b \):

Plot the y-intercept point (0, b) on the y-axis
Write the slope as a fraction: \( m = \frac{\text{rise}}{\text{run}} \)
From the y-intercept, use the slope to find another point:
- Move up/down by the rise
- Move right by the run
Plot the second point
Draw a straight line through both points
Extend the line in both directions with arrows

Examples:

Example 1: Graph \( y = 2x + 1 \)

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

Step 2: Slope = 2 = \( \frac{2}{1} \) (rise 2, run 1)

Step 3: From (0, 1), go up 2 and right 1 → reach (1, 3)

Step 4: Draw line through (0, 1) and (1, 3)

Example 2: Graph \( y = -\frac{3}{4}x + 2 \)

Y-intercept: (0, 2)

Slope: \( -\frac{3}{4} \) → down 3, right 4

From (0, 2), go down 3 and right 4 → reach (4, -1)

Draw line through both points

5. Graph a Line from an Equation in Point-Slope Form

Point-Slope Form:

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

where \( m \) = slope and \( (x_1, y_1) \) = a point on the line

Steps to Graph:

Identify the point \( (x_1, y_1) \) from the equation and plot it
Identify the slope \( m \)
From the plotted point, use the slope to find another point
Draw a line through both points

Example:

Graph: \( y - 3 = 2(x - 1) \)

Point: \( (x_1, y_1) = (1, 3) \) → Plot (1, 3)

Slope: \( m = 2 = \frac{2}{1} \)

From (1, 3), go up 2 and right 1 → reach (2, 5)

Draw line through (1, 3) and (2, 5)

6. Write a Linear Equation from a Slope and Y-Intercept

Steps:

Use the slope-intercept form: \( y = mx + b \)
Substitute the given slope for m
Substitute the given y-intercept for b
Write the final equation

Examples:

Example 1: Write an equation with slope 3 and y-intercept 5.

\( m = 3 \), \( b = 5 \)

Equation: \( y = 3x + 5 \)

Example 2: Write an equation with slope \( -\frac{1}{2} \) and y-intercept -4.

Equation: \( y = -\frac{1}{2}x - 4 \)

Example 3: Write an equation with slope 0 and y-intercept 6.

Equation: \( y = 0x + 6 \) or simply \( y = 6 \) (horizontal line)

7. Write a Linear Equation from a Graph

Steps:

Identify the y-intercept (where line crosses y-axis) → This is b
Choose two clear points on the line
Calculate the slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Write the equation: \( y = mx + b \)

Example:

A line crosses the y-axis at (0, -2) and passes through (3, 4). Write its equation.

Step 1: Y-intercept \( b = -2 \)

Step 2: Use points (0, -2) and (3, 4)

Step 3: \( m = \frac{4 - (-2)}{3 - 0} = \frac{6}{3} = 2 \)

Step 4: Equation: \( y = 2x - 2 \)

8. Write a Linear Equation from a Slope and a Point

Method 1: Point-Slope Form (Then Convert)

Use point-slope form: \( y - y_1 = m(x - x_1) \)
Substitute the given slope and point
Simplify to slope-intercept form if needed

Method 2: Find b, Then Use y = mx + b

Substitute the point and slope into \( y = mx + b \)
Solve for b
Write the final equation with m and b

Examples:

Example 1: Write an equation with slope 4 passing through (2, 5).

Method 1: \( y - 5 = 4(x - 2) \)

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

\( y = 4x - 3 \)

Method 2: \( 5 = 4(2) + b \)

\( 5 = 8 + b \) → \( b = -3 \)

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

9. Write a Linear Equation from Two Points

Steps:

Find the slope using: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Choose one point and use point-slope form, OR
Substitute one point into \( y = mx + b \) to find b
Write the final equation

Example:

Write an equation through points (1, 3) and (4, 12).

Step 1: Find slope

\( m = \frac{12 - 3}{4 - 1} = \frac{9}{3} = 3 \)

Step 2: Use point (1, 3) to find b

\( 3 = 3(1) + b \)

\( 3 = 3 + b \) → \( b = 0 \)

Equation: \( y = 3x \)

10. Convert Standard Form to Slope-Intercept Form

Standard Form:

\( Ax + By = C \)

Steps to Convert to \( y = mx + b \):

Subtract the x-term from both sides (isolate By)
Divide everything by B (coefficient of y)
Simplify to get y by itself

Examples:

Example 1: Convert \( 3x + 2y = 12 \) to slope-intercept form.

Step 1: \( 2y = -3x + 12 \)

Step 2: \( y = \frac{-3x + 12}{2} \)

Step 3: \( y = -\frac{3}{2}x + 6 \)

Example 2: Convert \( 4x - y = 8 \) to slope-intercept form.

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

\( y = 4x - 8 \)

11. Graph a Line from an Equation in Standard Form

Method 1: Find Intercepts

Find x-intercept: Set y = 0, solve for x
Find y-intercept: Set x = 0, solve for y
Plot both intercepts
Draw a line through the two points

Method 2: Convert to Slope-Intercept Form

Convert \( Ax + By = C \) to \( y = mx + b \)
Graph using slope and y-intercept

Example:

Graph: \( 2x + 3y = 6 \)

X-intercept: Set y = 0

\( 2x + 3(0) = 6 \) → \( x = 3 \) → Point (3, 0)

Y-intercept: Set x = 0

\( 2(0) + 3y = 6 \) → \( y = 2 \) → Point (0, 2)

Plot (3, 0) and (0, 2), then draw line

12. Graph a Horizontal or Vertical Line

Horizontal Lines:

Equation: \( y = k \) (where k is a constant)

Slope = 0
All points have the same y-coordinate
Parallel to the x-axis
Example: \( y = 3 \) → horizontal line through (0, 3)

Vertical Lines:

Equation: \( x = h \) (where h is a constant)

Slope = undefined
All points have the same x-coordinate
Parallel to the y-axis
Example: \( x = -2 \) → vertical line through (-2, 0)

How to Graph:

For \( y = k \): Draw a horizontal line through the point (0, k)

For \( x = h \): Draw a vertical line through the point (h, 0)

13. Equations of Horizontal and Vertical Lines

Writing Equations:

Horizontal Line: If line passes through point (a, b), equation is \( y = b \)

Vertical Line: If line passes through point (a, b), equation is \( x = a \)

Examples:

Example 1: Write the equation of a horizontal line through (5, -3).

Y-coordinate is -3

Equation: \( y = -3 \)

Example 2: Write the equation of a vertical line through (4, 7).

X-coordinate is 4

Equation: \( x = 4 \)

Example 3: Write the equation of the x-axis.

X-axis is a horizontal line where y = 0

Equation: \( y = 0 \)

14. Slopes of Parallel and Perpendicular Lines

Parallel Lines:

Parallel lines have the SAME slope

If \( m_1 = m_2 \), then the lines are parallel

Example: \( y = 3x + 1 \) and \( y = 3x - 5 \) are parallel (both have slope 3)

Perpendicular Lines:

Perpendicular lines have slopes that are negative reciprocals

If \( m_1 \times m_2 = -1 \), then the lines are perpendicular

Or: \( m_2 = -\frac{1}{m_1} \)

Examples:

Example 1: Are \( y = 2x + 3 \) and \( y = -\frac{1}{2}x + 1 \) perpendicular?

Slopes: \( m_1 = 2 \), \( m_2 = -\frac{1}{2} \)

\( m_1 \times m_2 = 2 \times (-\frac{1}{2}) = -1 \) ✓

Yes, they are perpendicular

Example 2: Find the slope of a line parallel to \( y = -4x + 7 \).

Original slope = -4

Parallel slope = -4 (same)

Example 3: Find the slope of a line perpendicular to \( y = \frac{3}{4}x - 2 \).

Original slope = \( \frac{3}{4} \)

Perpendicular slope = \( -\frac{4}{3} \) (negative reciprocal)

Quick Reference: Linear Equations

Three Forms of Linear Equations:

Form	Equation	Use
Slope-Intercept	\( y = mx + b \)	Easy to graph, shows slope and y-intercept
Point-Slope	\( y - y_1 = m(x - x_1) \)	Given slope and a point
Standard	\( Ax + By = C \)	Easy to find intercepts

Special Lines:

Horizontal: \( y = k \) (slope = 0)
Vertical: \( x = h \) (undefined slope)
Parallel lines: Same slope
Perpendicular lines: Slopes are negative reciprocals (\( m_1 \cdot m_2 = -1 \))