Linear Equations (One Variable)
1. Core Idea
A linear equation in one variable has the form ax + b = c, with the variable appearing to the first power only.
Solving means isolating the variable by performing the same operation on both sides. Every step is reversible, which is why you can always check your answer by substitution.
2. Must-Know Rules
| Step | Rule |
|---|---|
| Clear fractions | Multiply both sides by the LCM of denominators |
| Clear brackets | Distribute, watching signs |
| Collect | Variables to one side, constants to the other |
| Isolate | Divide by the coefficient |
| Check | Substitute back |
Number of solutions:
| Form after simplifying | Solutions |
|---|---|
| x = a specific number | exactly one |
| 0 = 0 (identity) | infinitely many |
| 0 = 5 (contradiction) | none |
Literal equations (solving for one variable in terms of others) use identical steps. To solve A = P(1 + rt) for r:
A/P = 1 + rt -> A/P - 1 = rt -> r = (A/P - 1)/t = (A - P)/(Pt).
Translation table for word problems:
| English | Algebra |
|---|---|
| is, was, will be | = |
| more than, sum, increased by | + |
| less than, difference, decreased by | - (mind the order!) |
| of, product, times | x |
| per, out of, ratio | / |
| what, a number | the variable |
Note: "5 less than x" is x - 5, not 5 - x.
3. Worked Examples
Example 1 — With fractions
Solve x/3 + 2 = x/5 + 6.
Multiply everything by 15:
5x + 30 = 3x + 90
2x = 60 -> x = 30.
Check: 30/3 + 2 = 12, and 30/5 + 6 = 12. Correct.
Example 2 — Brackets and negatives
Solve 4 - 2(x - 3) = 3x + 5.
Distribute carefully: 4 - 2x + 6 = 3x + 5
10 - 2x = 3x + 5
5 = 5x -> x = 1.
Example 3 — Word problem
The sum of three consecutive integers is 5 more than twice the smallest. Find the integers.
Let the smallest be n. Sum = n + (n+1) + (n+2) = 3n + 3.
3n + 3 = 2n + 5 -> n = 2.
Integers: 2, 3, 4. Check: sum 9; twice smallest plus 5 = 4 + 5 = 9. Correct.
4. GRE Traps
- Distributing a negative over only the first term. -2(x - 3) = -2x + 6.
- "5 less than x" written as 5 - x. Reverse the order.
- Multiplying both sides by an expression containing the variable without noting it could be zero — this can create or destroy solutions.
- Answering the wrong quantity. The question may ask for 2x + 1, not x. Underline what is asked.
- Forgetting to distribute to every term when clearing fractions.
- Assuming a solution exists. Some GRE questions are testing whether you notice "no solution".
5. Speed Tricks
- Clear fractions first. Working with integers is always faster than working with fractions.
- Move the smaller variable term to avoid negatives (in Example 2, moving -2x rather than 3x).
- Back-solve from the answer choices when the algebra looks ugly — start with choice C (the middle value) so one test tells you which direction to move.
- Check by substitution on any question worth more than 60 seconds; it costs 10 seconds and catches sign errors.
- For literal equations, treat every other letter as a number. The steps are unchanged.
6. Self-Check
Q1. Solve 3(x + 4) - 2 = 5x - 6.
Q2. Solve for h: V = (1/3)(pi)(r^2)h.
Q3. Seven less than three times a number is 26. What is the number?
Answers
A1. 3x + 12 - 2 = 5x - 6 -> 3x + 10 = 5x - 6 -> 16 = 2x -> x = 8.
A2. h = 3V / (pi r^2).
A3. 3n - 7 = 26 -> 3n = 33 -> n = 11.
7. One-Line Summary for the Board
Clear fractions, distribute carefully, collect, isolate, check. "Less than" reverses the order.