An equation is a balance.
Whatever you do to one side, do to the other.
Every legal move keeps the scale level. That is the only rule — everything below is a shortcut for applying it in a sensible order.
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- Grade 7-12 Math Review
- Linear Equations in One Variable
Linear Equations in One Variable
Undo the operations in reverse order, doing the same thing to both sides every time.
Solve for x
5x − 3 = 2x + 12
Subtract 2x from both sides: 3x − 3 = 12. Add 3: 3x = 15. Divide by 3: x = 5. Check in the original: 5(5) − 3 = 22 and 2(5) + 12 = 22 ✓ D stops at 3x = 15 and forgets the last division. C adds the x terms (5x + 2x = 7x) instead of subtracting, which is what moving a term across actually requires.
Two-step equations: plain numbers first.
Undo addition and subtraction before multiplication and division.
4x + 7 = 31
the equation
4x = 24
− 7 both sides
x = 6
÷ 4 both sides
This is the reverse of how the expression was built: x was multiplied by 4, then 7 was added, so the 7 comes off first.
Brackets: distribute, then carry on as normal.
3(2x − 5) = 21
brackets in the way
6x − 15 = 21
3 times both terms
6x = 36
+ 15
x = 6
÷ 6
Distribute to every term inside
3(2x − 5) is 6x − 15, not 6x − 5. And with a negative outside, −2(x − 4) is −2x + 8, because a negative times a negative is positive.
Here you could also divide both sides by 3 first, giving 2x − 5 = 7. Both routes reach x = 6; dividing first is quicker when the number divides cleanly.
Variables on both sides: gather them on one.
Move the smaller x term, and you avoid negatives.
5x − 3 = 2x + 12
the opening problem
3x − 3 = 12
− 2x both sides
3x = 15
+ 3
x = 5
÷ 3
Moving the 5x instead would give −3 = −3x + 12, then −3x = −15 and x = 5 — the same answer with two extra minus signs to get wrong.
Equations with a fraction: multiply to clear it.
Dividing x by a number is undone by multiplying both whole sides by that number.
Solve x5 − 2 = 4.
x5 = 6
add 2 first: plain numbers come off before the division
x = 30
multiply both sides by 5
305 − 2 = 6 − 2 = 4 ✓
check in the original
Multiplying only one term
Multiplying by 5 straight away but forgetting the −2 gives x − 2 = 20 and x = 22. Then 225 − 2 = 2.4, not 4. If you multiply first, every term on both sides is multiplied: x − 10 = 20, so x = 30.
When a whole expression sits over the number: solve 3x − 42 = 7.
3x − 4 = 14
multiply both sides by 2; the bar acts as a bracket
3x = 18
add 4
x = 6
check: 18 − 42 = 7 ✓
Variations: a minus before a bracket, a fraction answer, and the equations that break.
The same balance moves; these are the endings students do not expect.
Solve 10 − 3(x + 2) = 13.
10 − 3x − 6 = 13
−3 times both terms: −3x and −6
4 − 3x = 13, so −3x = 9
collect, then subtract 4
x = −3
check: 10 − 3(−1) = 13 ✓
The answer can be negative; trust it once the check works.
Solve 4x + 3 = 13.
4x = 10
subtract 3
x = 104 = 52
check: 4(2.5) + 3 = 13 ✓
An answer does not have to be a whole number; leave it as a fraction in lowest terms or as 2.5.
Solve 2(x + 3) = 2x + 5, then 3(x − 1) = 3x − 3.
2x + 6 = 2x + 5 → 6 = 5
x vanishes and a false statement is left: no solution
3x − 3 = 3x − 3 → −3 = −3
x vanishes and a true statement is left: every number works
When the x terms cancel, the leftover statement decides: false means no solution, true means infinitely many.
Always substitute the answer back.
It takes ten seconds and catches every sign error.
Left side
5(5) − 3 = 22
using x = 5
Right side
2(5) + 12 = 22
same ✓
Test a wrong answer
x = 3: 12 vs 18
not equal ✗
Substitute into the original equation, not into a line you wrote halfway down. If the mistake happened on line two, checking against line three will not find it.
On the exam
Equation items usually carry one complication — a bracket, a fraction, or x on both sides — and the wrong choices are the answers from a dropped sign. The fastest route is the flow above, clearing fractions and brackets first; the 10-second check is to put your answer into the original equation and see both sides match.
Now you try
Practice problem
Solve for x
4(x + 2) = 3x + 15
Distribute first: 4x + 8 = 3x + 15. Subtract 3x: x + 8 = 15. Subtract 8: x = 7. Check: 4(9) = 36 and 3(7) + 15 = 36 ✓ C comes from distributing only to the x, giving 4x + 2 = 3x + 15. B adds the x terms instead of subtracting.
Hint: the 4 multiplies both the x and the 2. Clear the bracket before anything else, then the variable terms.