Math Steps guide
How to solve equations step by step: linear, quadratic and simultaneous
To solve an equation, do the same thing to both sides until the unknown is on its own. For a linear equation such as 2x + 3 = 11, undo the operations in reverse order: subtract 3 from both sides to get 2x = 8, then divide both sides by 2 to get x = 4. Quadratics need factorising or the quadratic formula, and simultaneous equations need substitution or elimination. Always check your answer by putting it back into the original equation.
The one rule: keep it balanced
An equation is like a set of scales. The two sides are equal, and they stay equal as long as you do exactly the same thing to both. You can:
- Add or subtract the same number or term on both sides.
- Multiply or divide both sides by the same number (never by zero).
- Expand brackets or simplify one side, which doesn't change its value.
Your goal is to get the letter, usually x, alone on one side. Work backwards: undo addition and subtraction first, then multiplication and division.
Solving linear equations
A linear equation has x but no x², such as 2x + 3 = 11. The steps:
- Expand any brackets. 3(x − 2) becomes 3x − 6.
- Clear fractions by multiplying every term by the common denominator.
- Collect the x terms on one side and the numbers on the other, by adding or subtracting on both sides.
- Divide by the number in front of x.
- Check by substituting your answer into the original equation.
Worked example
Solve 3(x − 2) = x + 8
Expand: 3x − 6 = x + 8
Subtract x from both sides: 2x − 6 = 8
Add 6 to both sides: 2x = 14
Divide both sides by 2: x = 7
Check: 3(7 − 2) = 15, and 7 + 8 = 15. ✓
Solving quadratic equations
A quadratic has an x² term. First rearrange it so one side is zero: ax² + bx + c = 0. Then pick a method:
| Method | Best when | Example |
|---|---|---|
| Factorising | The numbers are small and whole | x² − 5x + 6 = (x − 2)(x − 3) |
| Quadratic formula | It won't factorise neatly | 2x² + 3x − 1 = 0 |
| Completing the square | You need the turning point too | x² + 6x + 2 = (x + 3)² − 7 |
Factorising: find two numbers that multiply to c and add to b. For x² − 5x + 6 = 0 they are −2 and −3, so (x − 2)(x − 3) = 0. If two things multiply to make zero, one of them must be zero, so x = 2 or x = 3.
The quadratic formula works every time:
x = (−b ± √(b² − 4ac)) / 2a
The part under the square root, b² − 4ac, is the discriminant. If it's positive there are two solutions, if it's zero there is one, and if it's negative there are no real solutions.
Worked example
Solve 2x² + 3x − 1 = 0, so a = 2, b = 3, c = −1.
Discriminant: 3² − 4 × 2 × (−1) = 9 + 8 = 17
x = (−3 ± √17) / 4, so x ≈ 0.28 or x ≈ −1.78
Solving simultaneous equations
With two unknowns you need two equations. There are two main methods:
- Elimination: add or subtract the equations so one letter cancels out.
- Substitution: rearrange one equation for a letter and put it into the other.
Worked example
Solve 2x + 3y = 7 and x − y = 1
From the second equation, x = y + 1.
Substitute: 2(y + 1) + 3y = 7, so 5y + 2 = 7, so 5y = 5 and y = 1.
Then x = 1 + 1, so x = 2. Check: 2(2) + 3(1) = 7 ✓
Checking your steps with Math Steps
Math Steps shows the working, not just the answer, so you can see where you went wrong:
- Type the problem in the box, such as
2x + 3 = 11orx^2 - 5x + 6 = 0, and tap Solve. Use Math keys for symbols, or Paste. For simultaneous equations, separate them with a comma:2x + 3y = 7, x - y = 1. - Every step is explained, and the preview typesets your problem so you can spot typos before solving. Turn on Explain like I'm 12 for friendlier wording.
- Graph plots the equation, so you can see a quadratic's roots where the curve crosses the x-axis.
- Practice generates fresh problems by topic, with hints and full steps, and there's a daily puzzle with a streak.
- History keeps past problems; star the ones you want to revisit.
Solving with full working, the grapher, the formula sheet, practice and the daily puzzle are free, with no ads and no account. It runs on your device, so it works offline. Math Steps Pro ($3.99 once) adds Check my work, which marks your own working line by line, plus matrices and word problems.
Common mistakes
- Doing something to only one side, or to only one term on a side.
- Sign errors when moving terms or expanding −(x − 3), which is −x + 3.
- Dividing by x in a quadratic, which loses the solution x = 0. Factorise instead.
- Forgetting the ± in the quadratic formula, so you find only one answer.
- Not checking. Substituting back takes seconds and catches most slips.
Questions
What is the first step in solving an equation?
Simplify each side: expand brackets and collect like terms. Then move the x terms to one side and the numbers to the other, doing the same thing to both sides.
How do I know if a quadratic will factorise?
Look for two whole numbers that multiply to c and add to b (when a is 1). If you can't find them quickly, use the quadratic formula, which always works.
What does it mean if the discriminant is negative?
If b² − 4ac is less than zero, the quadratic has no real solutions: its graph never crosses the x-axis.
Should I use substitution or elimination?
Substitution is easiest when one equation already has a letter on its own, such as x = y + 1. Elimination is quicker when the same letter has matching or easy-to-match numbers in both equations.
How do I check my answer?
Put your answer back into the original equation. If both sides come out equal, it's right. For simultaneous equations, check both equations.
Is Math Steps free?
Yes. Step-by-step solving, Explain like I'm 12, the grapher, the formula sheet, practice and the daily puzzle are free with no account. Math Steps Pro ($3.99 once) adds Check my work, matrices and word problems.