How to solve a one-variable equation
TEAS 7 outline: Solve one-variable equations
Also covers HESI A2 Math (Basic algebra)
- Do the same thing to both sides to keep the equation balanced.
- Undo each step with its inverse: subtract to undo adding, divide to undo multiplying.
- Simplify first, then undo adding or subtracting, then multiplying or dividing.
- Clear a fraction like 2/3 by multiplying both sides by 3/2.
- Check your answer in the original equation.
Swipe the picture sideways to see all of it.
An equation is like a balanced scale: the left side, 4 times x minus 9, has exactly the same value as the right side, 27.
Key facts
- A solution is the value that makes the equation a true statement when you substitute it for the variable.
- Whatever you do to one side of an equation, you must do to the other side to keep it balanced.
- Undo an operation with its inverse: subtract to undo adding, add to undo subtracting, divide to undo multiplying, multiply to undo dividing.
Show all 6 key factsShow fewer
- Order of work: simplify each side, gather the variable terms on one side, gather the plain numbers on the other, then divide so the variable has a coefficient of 1.
- To undo a fraction in front of the variable, such as 2/3, multiply both sides by its reciprocal, 3/2.
- Always check by putting your answer into the original equation; both sides must come out equal.
Quick check1 question · 2 more at the end
Solve for n: n/4 − 3 = 5
Correct answer: D.
Undo the subtraction first: add 3 to both sides to get n/4 = 8. Then undo the division by multiplying both sides by 4, so n = 32. Check: 32/4 − 3 = 8 − 3 = 5, which matches the right side.
A This multiplies only the n/4 term by 4 (n − 3 = 20), so n = 23; multiplying must apply to every term on both sides.
B This subtracts 3 from 5 instead of adding it (n/4 = 2), then multiplies by 4 to get 8.
C This correctly gets n/4 = 8 but then divides by 4 instead of multiplying, which gives 2.
D Correct: n/4 − 3 = 5 → n/4 = 8 → n = 32.
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: To solve a one-variable equation, get the variable alone. First simplify each side by distributing and combining like terms. Then use inverse operations, always doing the same thing to both sides: undo addition or subtraction first, then multiplication or division. Finally, substitute your answer into the original equation and check that both sides match.
1 What does it mean to solve an equation?Solving an equation means finding the one value that keeps both sides equal. 1 min
Solving an equation means finding the one value that keeps both sides equal. That value is the solution: put it back in for the variable (the letter), and both sides come out the same. You never need to "see" the answer; you just repeat one rule.
The one rule: do the same thing to both sides, and use the inverse (opposite) operation to peel away whatever is attached to the variable.
| If the variable has… | Undo it by… |
|---|---|
| + a number | subtracting that number from both sides |
| − a number | adding that number to both sides |
| × a number | dividing both sides by that number |
| ÷ a number | multiplying both sides by that number |
Order of work: simplify each side, undo adding or subtracting, undo multiplying or dividing, then check.
2 How do you solve a two-step equation?Undo the adding or subtracting first, then undo the multiplying or dividing. 1 min
Undo the adding or subtracting first, then undo the multiplying or dividing. Example: a pharmacy technician has four full blister cards; after 9 tablets are used, 27 are left. With x tablets per card, 4x − 9 = 27.
- Add 9 to both sides: 4x = 36.
- Divide both sides by 4: x = 9.
- Check in the original: 4(9) − 9 = 36 − 9 = 27. It matches, so each card holds 9 tablets.
Classic wrong turn: subtracting 9 instead of adding it, which gives 4x = 18 and x = 4.5. Ask, "What is being done to x, and what undoes it?"
3 What if there are parentheses and x on both sides?Clear the parentheses first, then move all the x terms to one side. 1 min
Clear the parentheses first, then move all the x terms to one side. Example: 3(x + 4) = 5x − 8.
- Distribute the 3 to both terms: 3x + 12 = 5x − 8.
- Keep x on the side with the bigger x number (5x) so it stays positive. Subtract 3x from both sides: 12 = 2x − 8.
- Add 8 to both sides: 20 = 2x.
- Divide both sides by 2: x = 10.
- Check: 3(10 + 4) = 42 and 5(10) − 8 = 42. Both sides match.
Classic wrong turn: writing 3x + 4. The number outside the parentheses multiplies every term inside; that slip leads to the wrong answer x = 6.
4 What if there is a fraction in front of x?Multiply both sides by the reciprocal of the fraction to get x alone. 1 min
Multiply both sides by the reciprocal of the fraction to get x alone. Example: two-thirds of a full bottle of hand sanitizer plus a 5-ounce refill makes 17 ounces, so (2/3)x + 5 = 17.
- Subtract 5 from both sides: (2/3)x = 12.
- Multiply both sides by the reciprocal, 3/2. A number times its reciprocal is 1, so x = 12 × 3/2.
- Mental math: 12 ÷ 2 = 6, then 6 × 3 = 18. A full bottle holds 18 ounces.
- Check: (2/3)(18) = 12, and 12 + 5 = 17.
Classic wrong turn: multiplying by 2/3 instead of 3/2, which gives 8. Sense check: two-thirds of the bottle is 12 ounces, so the whole bottle must be more than 12.
5 When should you use the calculator?The calculator does the arithmetic, but you still decide every algebra step. 1 min
The calculator does the arithmetic, but you still decide every algebra step. The TEAS has a basic on-screen calculator, not a scientific or graphing one.
| Situation | Best tool |
|---|---|
| Small whole numbers (36 ÷ 4, 12 × 3/2) | Mental math, it is faster |
| Decimals (0.15x = 7.80 → 7.80 ÷ 0.15 = 52) | Calculator |
| Checking your answer (0.15 × 52 = 7.8) | Calculator |
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: Distribute to every term inside the parentheses: 3(x + 4) is 3x + 12, not 3x + 4.
Watch out: Use the inverse operation, not the same one: to remove − 9, add 9 to both sides; subtracting 9 makes it worse.
Watch out: Undo adding and subtracting before multiplying and dividing; dividing first only works if you divide every single term.
Watch out: To clear 2/3 from (2/3)x, multiply by 3/2, not by 2/3.
Watch out: Check in the original equation, not in a line you already rewrote, or you may just repeat your own mistake.
How to remember it
Undo the steps in the reverse order they were done to x. Think "last on, first off": socks go on before shoes, but shoes come off first. In 4x − 9, x was multiplied and then 9 was subtracted, so undo the subtraction first.
Self-check:
- Solve 5x + 6 = 41 and say which operation you undid first and why.
- Rewrite 2(x − 7) correctly before solving anything.
- Plug your answer from step 1 into the original equation to prove it works.
Show answers
- x = 7. Undo the + 6 first.
- x was multiplied by 5, then 6 was added. Last on, first off.
- Subtract 6 from both sides: 5x = 35.
- Divide both sides by 5: x = 7.
- 2x − 14.
- The 2 multiplies every term inside: 2 × x = 2x and 2 × 7 = 14.
- It works.
- Put 7 into the original: 5(7) + 6 = 35 + 6 = 41.
- Both sides equal 41.