Introduction to Linear Equations
Grade 7 · Math · Study Guide
Learn to solve one- and two-step linear equations using inverse operations and properties of equality.
A linear equation is a mathematical statement that two expressions are equal, involving a variable (usually x) raised only to the first power. For example, 2x + 5 = 11 is a linear equation.
To solve a linear equation, use inverse operations to isolate the variable. Inverse operations undo each other: addition and subtraction are inverse operations, and so are multiplication and division.
The Properties of Equality state that whatever you do to one side of an equation, you must do to the other side to keep it balanced. Think of an equation like a balance scale — both sides must stay equal.
For a one-step equation like x + 7 = 12, subtract 7 from both sides to get x = 5. For a two-step equation like 3x - 4 = 11, first add 4 to both sides (3x = 15), then divide both sides by 3 (x = 5).
Always check your answer by substituting it back into the original equation. Solutions to linear equations can be positive, negative, or fractional numbers. Writing equations from word problems is an important skill: look for key words like "sum," "difference," "times," and "divided by" to translate words into mathematical symbols.
Key terms
- Linear equation
- An equation where the variable has an exponent of 1; its graph is a straight line
- Variable
- A letter that represents an unknown number in an equation
- Inverse operations
- Operations that undo each other: addition/subtraction and multiplication/division
- Properties of Equality
- Whatever operation is performed on one side of an equation must be performed on the other side
- Solution
- The value of the variable that makes the equation true
- Isolate the variable
- Use inverse operations to get the variable alone on one side of the equation
Flashcards
- One-step equation
- An equation that requires only one inverse operation to solve; e.g., x + 8 = 15
- Two-step equation
- An equation that requires two inverse operations to solve; e.g., 2x + 3 = 11
- Solving order for two-step equations
- First undo addition or subtraction; then undo multiplication or division
- Addition Property of Equality
- Adding the same number to both sides of an equation keeps it balanced
- Subtraction Property of Equality
- Subtracting the same number from both sides of an equation keeps it balanced
- Multiplication Property of Equality
- Multiplying both sides by the same non-zero number keeps the equation balanced
- Division Property of Equality
- Dividing both sides by the same non-zero number keeps the equation balanced
- Checking a solution
- Substitute the answer back into the original equation; both sides should simplify to the same value
Questions
What is the solution to 4x - 6 = 18?
- x = 3
- x = 4
- x = 6
- x = 8
Show answer
Answer: x = 6
Add 6 to both sides: 4x = 24. Then divide by 4: x = 6. Check: 4(6) - 6 = 24 - 6 = 18 ✓.
To solve 5x = 35, you divide both sides by 5.
Show answer
Answer: True
Division is the inverse of multiplication. Dividing both sides by 5 gives x = 7.
To check a solution, you substitute the value back into the _______ equation.
Show answer
Answer: original
Substituting your answer back into the original equation verifies whether it is correct.
A number decreased by 9 equals 14. Which equation represents this situation?
- n + 9 = 14
- n - 9 = 14
- 9 - n = 14
- n × 9 = 14
Show answer
Answer: n - 9 = 14
'A number decreased by 9' means n - 9. 'Equals 14' means = 14. So the equation is n - 9 = 14.
The equation x² + 3 = 12 is a linear equation.
Show answer
Answer: False
A linear equation has the variable raised only to the first power. In x² + 3 = 12, x is squared (exponent 2), making it a quadratic equation, not linear.
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