A linear equation has a variable raised only to the first power, like 3x + 7 = 22 or 5(x − 2) = 3x + 4. Solving one means finding the value of x that makes both sides equal. Every method below rests on one rule: whatever you do to one side, do to the other.
The goal: get x alone
Think of the equation as a balance. To isolate x, undo what has been done to it, in reverse order. If x was multiplied by 3 and then 7 was added, subtract 7 first and divide by 3 second.
Example Solve 3x + 7 = 22
- Subtract 7 from both sides: 3x = 15.
- Divide both sides by 3: x = 5.
- Check by substituting: 3(5) + 7 = 15 + 7 = 22. ✓
The check takes ten seconds and catches most arithmetic slips. Make it a habit on every problem you can.
Multi-step equations
Longer equations follow the same idea with a fixed order:
- Clear parentheses with the distributive property.
- Combine like terms on each side.
- Move the variable terms to one side and the numbers to the other.
- Divide by the coefficient of x.
- Check.
Example Solve 5(x − 2) = 3x + 4
- Distribute: 5x − 10 = 3x + 4.
- Subtract 3x from both sides: 2x − 10 = 4.
- Add 10 to both sides: 2x = 14.
- Divide by 2: x = 7.
- Check: 5(7 − 2) = 25 and 3(7) + 4 = 25. ✓
When x appears on both sides, move it to the side where its coefficient stays positive. In the example above, subtracting 3x leaves 2x on the left. Subtracting 5x instead would work too, but you would end up dividing by a negative number, which is one more place to drop a sign.
Example Solve 4 − 2x = 10
- Subtract 4 from both sides: −2x = 6.
- Divide by −2: x = −3.
- Check: 4 − 2(−3) = 4 + 6 = 10. ✓
Distribute to every term
5(x − 2) is 5x − 10. Writing 5x − 2 is the most common error in multi-step equations. A minus sign in front of parentheses distributes too: −(x − 4) is −x + 4.
Equations with fractions
Fractions are easier to remove than to work with. Multiply every term on both sides by the least common denominator (LCD) and the fractions disappear.
Example Solve x/3 + x/4 = 7
- The LCD of 3 and 4 is 12. Multiply every term by 12: 4x + 3x = 84.
- Combine: 7x = 84.
- Divide: x = 12.
- Check: 12/3 + 12/4 = 4 + 3 = 7. ✓
When each side is a single fraction, cross-multiply. For (x + 2)/5 = (x − 1)/2, you get 2(x + 2) = 5(x − 1).
Equations with decimals
Decimals work the same way. Multiply by 10, 100 or 1,000 to make every number whole.
Example Solve 0.4x + 1.2 = 3
- Multiply every term by 10: 4x + 12 = 30.
- Subtract 12: 4x = 18.
- Divide by 4: x = 4.5.
When there is no solution, or every number works
Sometimes the x terms cancel. What is left tells you the answer.
- If you end with a false statement like 6 = 5, there is no solution.
- If you end with a true statement like −3 = −3, every real number is a solution.
Example Solve 2(x + 3) = 2x + 5
Distribute: 2x + 6 = 2x + 5. Subtract 2x: 6 = 5. That is false, so no value of x works. The equation has no solution.
Example Solve 3(x − 1) = 3x − 3
Distribute: 3x − 3 = 3x − 3. Both sides are identical, so the equation is true for every x. It has infinitely many solutions.
The SAT asks about this directly: “For what value of k does the equation have no solution?” Set the x coefficients equal and make the constants different.
Solving for a variable in a formula
A literal equation has several letters. You solve it for one of them using exactly the same steps, treating the other letters as numbers.
Example Solve A = ½bh for h
- Multiply both sides by 2: 2A = bh.
- Divide both sides by b: h = 2A/b.
Example Solve y = mx + b for x
- Subtract b: y − b = mx.
- Divide by m: x = (y − b)/m.
This is how you turn a formula you know into the one you need, for example getting time from distance = rate × time.
Word problems
Most word problems reduce to a linear equation once you name the unknown. Write down what x stands for before you write anything else.
Example A phone plan costs $25 a month plus $0.10 per minute. One month’s bill was $40. How many minutes were used?
- Let m be the number of minutes.
- The cost is 25 + 0.10m, so 25 + 0.10m = 40.
- Subtract 25: 0.10m = 15.
- Divide by 0.10: m = 150 minutes.
Example Sam has $120 and saves $15 a week. Lee has $60 and saves $25 a week. After how many weeks will they have the same amount?
- Let w be the number of weeks. Sam has 120 + 15w and Lee has 60 + 25w.
- Set them equal: 120 + 15w = 60 + 25w.
- Subtract 15w and 60: 60 = 10w, so w = 6.
- Check: after 6 weeks each has $210. ✓
Try a few
Try each one before opening the answer.
Solve 7x − 4 = 31.
Show answer
x = 5. Add 4 to get 7x = 35, then divide by 7.
Solve 2(3x + 1) = 4x + 16.
Show answer
x = 7. Distribute to get 6x + 2 = 4x + 16, subtract 4x and 2 to get 2x = 14.
Solve (x + 2)/5 = (x − 1)/2.
Show answer
x = 3. Cross-multiply: 2(x + 2) = 5(x − 1), so 2x + 4 = 5x − 5, which gives 9 = 3x.
Solve 4(x − 3) − 2x = 2(x + 1).
Show answer
No solution. The left side simplifies to 2x − 12 and the right to 2x + 2. Subtracting 2x leaves −12 = 2, which is false.
Solve P = 2l + 2w for w.
Show answer
w = (P − 2l)/2. Subtract 2l from both sides, then divide by 2.
Solve 0.25x + 0.5 = 2.
Show answer
x = 6. Multiply every term by 4 to get x + 2 = 8.
A gym charges a $40 sign-up fee and $18 a month. After how many months will you have paid $256 in total?
Show answer
12 months. Solve 40 + 18m = 256: 18m = 216, so m = 12.
Practice
Questions students ask
Do I have to check my answer?
On homework, yes, at least until you stop making sign errors. On a timed test, check the problems where you distributed a negative or cleared fractions, since those are where mistakes happen.
Which side should x end up on?
It doesn’t matter. x = 5 and 5 = x mean the same thing. Pick the side that keeps the coefficient positive.
What is the difference between an expression and an equation?
An expression, like 3x + 7, has no equals sign and can only be simplified. An equation sets two expressions equal and can be solved.
Next, see how linear equations look on a graph in Slope and Linear Equations, or solve two at once in Systems of Equations.