Linear Equations
Grade 9 · Math · Study Guide
Solve one-variable linear equations and understand what solutions mean in real-world contexts.
A linear equation is any equation where the variable appears to the first power — no squares, no roots. The goal is always to isolate the variable by performing the same operation on both sides. For example, to solve 3x + 7 = 22, subtract 7 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5.
Multi-step equations may involve combining like terms or using the distributive property first. For 2(x + 4) = 18, distribute to get 2x + 8 = 18, subtract 8 to get 2x = 10, then divide to get x = 5. Always check your answer by substituting back into the original equation.
Some equations have variables on both sides. Move all variable terms to one side and constants to the other. For 5x − 3 = 2x + 9, subtract 2x to get 3x − 3 = 9, then add 3 to get 3x = 12, so x = 4. Special cases: an equation that simplifies to a true statement (like 0 = 0) has infinitely many solutions; one that simplifies to a false statement (like 5 = 0) has no solution.
| Step | Action | Example |
|---|---|---|
| 1 | Distribute / combine like terms | 2(x+3)+x = 5x becomes 2x+6+x = 3x+6 |
| 2 | Move variables to one side | 3x+6 = 12 → subtract 6 |
| 3 | Move constants to other side | 3x = 6 |
| 4 | Divide by coefficient | x = 2 |
| 5 | Check by substituting | 2(2+3)+2 = 12 ✓ |
Key terms
- Linear equation
- An equation where the variable is raised only to the first power
- Solution
- The value of the variable that makes the equation true
- Inverse operations
- Operations that undo each other, such as addition and subtraction
- Distributive property
- a(b + c) = ab + ac — multiply the outside factor by each term inside
- Identity
- An equation true for all values of the variable; infinitely many solutions
- Contradiction
- An equation that is never true; no solution
Flashcards
- Linear equation
- An equation where the highest power of the variable is 1
- Solution of an equation
- A value that makes the equation true when substituted for the variable
- Inverse operations
- Pairs like add/subtract or multiply/divide used to isolate a variable
- Distributive property
- a(b + c) = ab + ac
- Like terms
- Terms with the same variable and exponent; they can be combined
- Identity equation
- Simplifies to 0 = 0; true for all values — infinite solutions
Practice questions
Solve: 4x − 5 = 19
- x = 3
- x = 6
- x = 4
- x = 7
Show answer
Answer: x = 6
Add 5 to both sides: 4x = 24; divide by 4: x = 6.
Solve: 3(x + 2) = 21
- x = 5
- x = 7
- x = 9
- x = 6
Show answer
Answer: x = 5
Distribute: 3x + 6 = 21; subtract 6: 3x = 15; divide by 3: x = 5.
The equation 2x + 4 = 2(x + 2) has infinitely many solutions.
Show answer
Answer: True
Expanding the right side gives 2x + 4 = 2x + 4, which is always true.
If 7x = 49, then x = ___.
Show answer
Answer: 7
Divide both sides by 7: x = 49 ÷ 7 = 7.
Solve: 5x − 3 = 2x + 9
- x = 2
- x = 4
- x = 6
- x = 3
Show answer
Answer: x = 4
Subtract 2x from both sides: 3x − 3 = 9; add 3: 3x = 12; divide by 3: x = 4.
Fun facts
- The word 'algebra' comes from the Arabic word 'al-jabr,' meaning 'the reunion of broken parts' — literally putting equations back together.
- Linear equations model countless real situations: distance = rate × time, total cost = price × quantity, and many more.
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