Solving Linear Equations

Grade 8 · Math · Study Guide

Learn how to solve one- and two-step linear equations using inverse operations.

A linear equation is a mathematical statement that says two expressions are equal, and it involves a variable raised to the first power — no exponents like x² here. The goal is always the same: isolate the variable on one side of the equals sign by using inverse operations.

Inverse operations are opposites. Addition and subtraction are inverses of each other, and so are multiplication and division. Whatever you do to one side of the equation, you must do to the other side to keep it balanced — think of an equation like a scale.

For a two-step equation like 3x + 5 = 20, you work in reverse order of operations. First, subtract 5 from both sides to get 3x = 15. Then divide both sides by 3 to find x = 5. Always check your answer by substituting it back into the original equation.

Linear equations model real-world situations constantly. If a taxi charges $2 per mile plus a $5 base fee, and your bill is $21, you can write 2x + 5 = 21 and solve for x to find you traveled 8 miles. This is why linear equations are one of the most powerful tools in algebra.

Key terms

Linear Equation
An equation where the variable is raised to the first power, producing a straight line when graphed.
Inverse Operations
Operations that undo each other: addition/subtraction and multiplication/division are pairs of inverse operations.
Variable
A letter (like x or y) that represents an unknown value.
Coefficient
The number multiplied by the variable in a term, such as the 3 in 3x.
Solution
The value of the variable that makes the equation true.

Flashcards

Linear Equation
An equation with a variable to the first power; its graph is a straight line.
Inverse Operations
Pairs that undo each other: + and −, × and ÷.
Isolate the Variable
Use inverse operations to get the variable alone on one side of the equation.
Coefficient
The number in front of a variable (the 4 in 4x).
Constant
A fixed number in an equation with no variable attached.
Two-Step Equation
An equation that requires two inverse operations to solve (e.g., 3x + 5 = 20).
Solution
The value that makes both sides of the equation equal.

Questions

  1. Solve for x: 2x + 6 = 14

    1. x = 4
    2. x = 10
    3. x = 3
    4. x = 7
    Show answer

    Answer: x = 4

    Subtract 6 from both sides: 2x = 8. Divide both sides by 2: x = 4.

  2. In the equation 5x − 3 = 12, the first step is to add 3 to both sides.

    Show answer

    Answer: True

    To undo subtracting 3, you add 3 to both sides first, giving 5x = 15, then divide by 5.

  3. Which value of x satisfies the equation x/4 − 2 = 3?

    1. x = 5
    2. x = 20
    3. x = 10
    4. x = 4
    Show answer

    Answer: x = 20

    Add 2 to both sides: x/4 = 5. Multiply both sides by 4: x = 20.

  4. To check a solution, substitute the value back into the ___ equation.

    Show answer

    Answer: original

    Substituting the answer back into the original equation confirms whether both sides are equal.

  5. A number is multiplied by 7, then 3 is subtracted, giving 25. Which equation represents this?

    1. 7n + 3 = 25
    2. 7n − 3 = 25
    3. 3n − 7 = 25
    4. n/7 − 3 = 25
    Show answer

    Answer: 7n − 3 = 25

    'Multiplied by 7' means 7n; 'then 3 is subtracted' means − 3; 'giving 25' means = 25. So 7n − 3 = 25.

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