Example 1: Basic Linear Equation

Problem: Solve 5x - 8 = 17

Solution:
Add 8 to both sides: 5x = 25
Divide by 5: x = 5
Check: 5(5) - 8 = 25 - 8 = 17 ✓

Example 2: Equation with Variables on Both Sides

Problem: Solve 7x + 3 = 4x + 15

Solution:
Subtract 4x from both sides: 3x + 3 = 15
Subtract 3 from both sides: 3x = 12
Divide by 3: x = 4

Example 3: Real-World Application

Problem: A taxi charges $3 plus $0.50 per mile. If your fare was $15.50, how many miles did you travel?

Solution: Set up the equation: 3 + 0.50m = 15.50 (where m is miles)
Subtract 3: 0.50m = 12.50
Divide by 0.50: m = 25 miles

This demonstrates how linear equations model real-world situations: a fixed charge (y-intercept) plus a variable rate (slope).

Example 4: Graphing y = 2x + 3

Problem: Graph the linear equation y = 2x + 3

Solution:
The slope m = 2 means rise 2, run 1 (or up 2 units for every 1 unit right).
The y-intercept b = 3 means the line crosses the y-axis at (0, 3).
Start at (0, 3), then move right 1 and up 2 to get (1, 5), then to (2, 7), etc.
Connect these points with a straight line.

Always Isolate the Variable

The goal in solving is to get the variable alone on one side of the equation. Use inverse operations systematically: if something is added to x, subtract it from both sides; if x is multiplied by a number, divide both sides by that number. Work step-by-step, undoing operations in reverse order of the order of operations (PEMDAS backwards).

Maintain Balance

The fundamental rule of equation solving: whatever you do to one side, do to the other. This keeps the equation balanced and the equality valid. If you add 5 to the left side, add 5 to the right side. If you divide the left by 3, divide the right by 3. This principle ensures your solution is correct.

Check Your Solutions

Always verify your answer by substituting it back into the original equation. If x = 7 is your solution, plug 7 into the original equation and confirm both sides are equal. This simple check catches arithmetic errors immediately and confirms you solved correctly. Make checking a habit—it takes seconds and prevents mistakes.

Recognize the Form y = mx + b

When a linear equation is in slope-intercept form, you can immediately identify the slope and y-intercept. This makes graphing easy and helps you understand the relationship between variables. If an equation isn't in this form, rearrange it by solving for y to reveal the slope and intercept.

Linear Equation

An equation where the variable appears only to the first power (no squares, cubes, or other exponents). The general form is ax + b = c. Linear equations graph as straight lines and represent proportional relationships.

Slope (m)

The rate of change in a linear relationship, calculated as rise over run: m = (change in y)/(change in x). Slope tells you how steep the line is and whether it rises (positive), falls (negative), or is horizontal (zero). It's the coefficient of x in y = mx + b.

Y-Intercept (b)

The point where a line crosses the y-axis, occurring when x = 0. In the equation y = mx + b, b is the y-intercept. It represents the starting value or initial condition in many real-world applications.

Variable

A symbol (usually a letter like x or y) that represents an unknown value. In linear equations, we solve to find the value of the variable that makes the equation true. Variables can represent any quantity—distance, time, cost, temperature, etc.

Coefficient

The number multiplied by the variable in an algebraic term. In 5x, the coefficient is 5. In y = mx + b, m is the coefficient of x. Coefficients tell you how many times to count the variable.

Solution

The value(s) that satisfy an equation—that make it true. For the equation 2x + 1 = 7, the solution is x = 3 because substituting 3 gives 2(3) + 1 = 7, which is true. Linear equations typically have exactly one solution, no solution, or infinitely many solutions.

Slope-Intercept Form

The form y = mx + b, where m is the slope and b is the y-intercept. This is the most useful form for graphing and understanding linear relationships because it immediately shows the line's steepness and starting point.