Skip to content

Linear Equations

Definition

A linear equation is a mathematical statement that contains variables where the highest exponent of any variable is 1. In a two-variable linear equation, the graph forms a straight line, hence the name "linear."

The standard form of a linear equation is: 

ax + by = c
Where a, b, and c are constants, and at least one of a or b is not zero.

Prerequisites

To understand and solve linear equations, you should have mastered:

  1. Variables and Expressions
  2. Order of Operations
  3. Basic Equation Solving
  4. Coordinate Plane (for graphing)

Learning Objectives

After mastering linear equations, you should be able to:

  1. Identify linear equations in various forms (standard, slope-intercept, point-slope)
  2. Convert between different forms of linear equations
  3. Solve linear equations for a specific variable
  4. Graph linear equations on a coordinate plane
  5. Interpret the slope and y-intercept of a linear equation
  6. Construct linear equations to model real-world situations
  7. Apply linear equations to solve word problems

Forms of Linear Equations

Standard Form

ax + by = c

Slope-Intercept Form

y = mx + b
Where: - m is the slope (rate of change) - b is the y-intercept (where the line crosses the y-axis)

Point-Slope Form

y - y₁ = m(x - x₁)
Where: - (x₁, y₁) is a point on the line - m is the slope

Examples

Example 1: Basic Linear Equation (Early Middle School)

Problem: Solve for x: 2x + 5 = 11

Solution: 1. Subtract 5 from both sides:
2x + 5 - 5 = 11 - 5
2x = 6 2. Divide both sides by 2:
2x ÷ 2 = 6 ÷ 2
x = 3

Check: Replace x with 3 in the original equation:
2(3) + 5 = 6 + 5 = 11 ✓

Example 2: Converting Between Forms (Late Middle School)

Problem: Convert the standard form equation 3x - 2y = 12 to slope-intercept form.

Solution: 1. Solve for y:
-2y = 12 - 3x
-2y = -3x + 12
Divide all terms by -2:
y = (3/2)x - 6

Interpretation: The slope is 3/2, and the y-intercept is -6.

Example 3: Real-World Application (High School)

Problem: A cell phone plan costs $25 per month plus $0.10 per text message. Write a linear equation that models the monthly cost y as a function of the number of text messages x.

Solution: - Fixed cost = $25 - Variable cost = $0.10 per text message - Linear equation: y = 0.10x + 25

Interpretation: In slope-intercept form, the slope (0.10) represents the cost per text message, and the y-intercept (25) represents the fixed monthly cost.

Extension: How many text messages can be sent for a monthly bill of $40? 

40 = 0.10x + 25
40 - 25 = 0.10x
15 = 0.10x
15 ÷ 0.10 = x
150 = x
Answer: 150 text messages

Example 4: Graphing Linear Equations (High School)

Problem: Graph the line with the equation y = -2x + 4.

Solution: 1. Identify the slope and y-intercept: - Slope (m) = -2 - y-intercept (b) = 4 2. Plot the y-intercept at (0, 4) 3. From this point, use the slope to find another point: - Slope = -2 = -2/1 means go down 2 and right 1 - This gives the point (1, 2) 4. Draw a line through these two points.

Common Misconceptions

  1. Confusing the x and y-intercepts: The x-intercept occurs where y = 0, while the y-intercept occurs where x = 0.
  2. Misinterpreting the slope: A negative slope means the line decreases as x increases, not that the line goes from right to left.
  3. Forgetting to distribute: When solving equations like 2(x + 3) = 10, students often forget to multiply both terms inside the parentheses by 2.

Applications in SAT

Linear equations appear throughout the SAT Math section, particularly in:

  1. Heart of Algebra questions: These may ask you to solve linear equations or interpret their graphs.
  2. Word problems: Many real-world scenarios are modeled with linear equations.
  3. Systems of equations: Linear equations are combined to solve for multiple variables.

Advanced Connections

Linear equations connect to:

  1. Linear Inequalities - Using similar skills but with inequality symbols
  2. Systems of Equations - Solving multiple linear equations simultaneously
  3. Functions - Linear equations are one type of function

Practice Problems

  1. Basic: Solve for x: 5x - 3 = 27
  2. Intermediate: Convert 2x + 3y = 12 to slope-intercept form and identify the slope and y-intercept.
  3. Advanced: A company's profit P (in dollars) after selling n items is modeled by the equation P = 25n - 1000. How many items must be sold to break even (P = 0)?
  4. SAT-Level: The equation of a line in the xy-plane is y = -3x + b. The line passes through the point (4, -8). What is the value of b?