Key takeaways
- An equation is a balance: do the same thing to both sides and it stays true.
- To solve for x, undo operations in reverse order until x is alone.
- Two unknowns need two equations. Add, subtract or substitute to remove one unknown.
- Quadratics (ax² + bx + c = 0) can have two, one or no real solutions.
- Always check your answer by putting it back into the original equation.
In this guide
What is algebra?
Algebra is maths with letters. A letter such as x stands for a number we don't know yet. Algebra gives us rules to find that number.
An expression is a group of numbers and letters, like 3x + 5. An equation says two expressions are equal, like 3x + 5 = 20. Solving the equation means finding the value of x that makes it true.
Why does algebra matter?
Algebra is the language behind almost every formula you'll ever use: interest on a loan, speed and distance, the size of a phone screen, the price of a taxi ride. When you know how to rearrange a formula, one formula can answer many different questions.
If a taxi charges $3 plus $2 per km and the fare was $19, how far did you travel? That is 3 + 2x = 19, and algebra says x = 8 km.The main types of equations
| Type | General form | Example | Number of solutions |
|---|---|---|---|
| Linear | ax + b = c | 3x + 5 = 20 | Usually one |
| Simultaneous | two equations, two unknowns | x + y = 10, x − y = 2 | Usually one pair |
| Quadratic | ax² + bx + c = 0 | x² − 5x + 6 = 0 | Two, one or none (real) |
For quadratics, the key tool is the quadratic formula:
How to solve a linear equation step by step
- 1Simplify each side. Expand brackets and collect like terms.
- 2Move the x terms to one side. Add or subtract the same x term on both sides.
- 3Move the plain numbers to the other side. Add or subtract on both sides.
- 4Divide by the coefficient so that x is alone.
- 5Check. Put your answer back into the original equation.
The linear equation solver shows the same balancing steps for any equation of this type.
Two equations, two unknowns
Suppose x + y = 10 and x − y = 2. Add the two equations and the y terms cancel: 2x = 12, so x = 6. Put that back: 6 + y = 10, so y = 4. This is called the elimination method. You can check pairs like this with the simultaneous equations calculator.
Quadratic equations
For x² − 5x + 6 = 0, we have a = 1, b = −5 and c = 6. The part under the square root is (−5)² − 4 × 1 × 6 = 1. So x = (5 ± 1) ÷ 2, which gives x = 3 or x = 2. The quadratic formula calculator works through this for any a, b and c.
Common mistakes to avoid
- Sign slips. Moving
+5to the other side makes it−5. Write each step so you can see the change. - Dividing only part of a side. In
2x + 4 = 10, don't divide just2xby 2. Subtract 4 first, or divide every term. - Squaring a negative wrongly.
(−5)² = 25, but−5²means−25. Use brackets. - Forgetting the second root of a quadratic. The ± gives two answers.
Tips and tricks
- Checking is free marks. Substituting the answer back takes seconds and catches most slips.
- Try factoring first. If a quadratic factors easily, like
(x − 2)(x − 3) = 0, it is faster than the formula. - Straight lines are algebra too. The slope calculator finds how steep a line is from two points.
Where you'll use it in real life
- Money: working out how many months to save for something, or comparing phone plans.
- Travel: rearranging
distance = speed × timeto find the time. - Business: finding the break-even point where costs equal sales.
- Science: rearranging formulas like
F = mato find the mass.
Quick summary
Algebra is about finding unknown numbers while keeping equations balanced. For linear equations, undo each operation on both sides until x is alone. For two unknowns, eliminate one. For quadratics, use factoring or the quadratic formula. Always check your answer, and use the free solvers linked above to see every step.
Frequently asked questions
How do I solve for x?
Get x alone by doing the same operation to both sides of the equation. Undo additions and subtractions first, then multiplications and divisions.
What does the quadratic formula solve?
It solves any equation of the form ax² + bx + c = 0, giving the x-values where the parabola crosses the x-axis.
What is the difference between an expression and an equation?
An expression, like 3x + 5, has no equals sign and cannot be solved. An equation, like 3x + 5 = 20, says two things are equal, so you can solve it for x.
How can I check my answer?
Put your value of x back into the original equation. If both sides give the same number, your answer is correct.
