Algebra Lessons: How to Solve an Algebra Problem

Every algebra problem comes down to a few moves: simplify, do the same thing to both sides, and check. These algebra lessons teach each move with worked examples whose answers are checked by Sumzo’s math engine.

18 articles

5 min read
Equations: what they are and how to solve them
What an equation is, what its solution means, and how to solve equations step by step — from one-step equations to fractions and two variables.
1 min read
Solving linear equations
Use inverse operations to get the unknown on its own, keep the equation balanced, and always check your answer.
2 min read
Quadratic equations: how to solve them
Factorising, completing the square, the quadratic formula and the discriminant — and how to tell which method to use.
1 min read
Slope and the equation of a line
What slope measures, how to find it from two points, and how to write a line as y = mx + b.
4 min read
How to simplify radicals (simplest radical form)
How to simplify radicals by taking out perfect-square factors, what simplest radical form means, and how to simplify radical expressions — with checked examples.
2 min read
Rational equations: how to solve them
What a rational equation is, how to solve rational equations by multiplying by the lowest common denominator, and why every answer must be checked.
3 min read
Factoring polynomials: a step-by-step guide
The steps for factoring polynomials — common factor first, then two, three or four terms — with worked examples, each checked by multiplying back out.
3 min read
Solving inequalities: how to solve and graph them
Solving inequalities like equations — with one extra rule — and showing the answer on a number line, in interval notation and as a shaded region.
2 min read
What is a solution in math? Examples with one, none or many
A solution is a value that makes an equation true. How to check one, and examples with one solution, no solution, all real numbers and two solutions.
2 min read
How to solve a word problem: 5 steps that work
A five-step method for solving word problems, the key words that turn into maths, and worked examples — ages, consecutive numbers, discounts and tickets.
2 min read
How to graph an equation: lines, parabolas and more
Three ways to graph an equation — a table of values, slope and intercept, and key points of a parabola — with worked examples and how to match a graph to its equation.
3 min read
Functions: how to graph a function, domain, range and f(g(x))
Function notation, how to graph a function from its formula, how to find its domain and range, and how composite functions such as f(g(x)) work.
2 min read
Polynomials: how to simplify, expand and solve them
What a polynomial is, how to simplify and expand polynomials, and how to solve polynomial equations by factoring — with checked examples.
2 min read
Rational expressions: simplify, multiply, divide, add and subtract
What rational expressions are, their excluded values, and how to simplify, multiply, divide, add and subtract them — then solve rational equations.
2 min read
How to solve literal equations (rearranging formulas)
A literal equation has several letters. Solving it means isolating one of them — the same moves as an ordinary equation, with letters instead of numbers.
1 min read
How to simplify expressions and complex fractions
How to simplify the following kinds of expression: like terms, brackets, exponents and complex fractions (a fraction inside a fraction), each with checked examples.
1 min read
Find x: how to solve for x in equations, angles and triangles
The questions that say “find x” — equations, angles on a line, triangle angles, Pythagoras and similar shapes — and the one idea behind all of them.
2 min read
Systems of equations and graphing systems of linear inequalities
How to find the solution to a system of equations three ways, then how to graph a system of inequalities: boundary lines, shading and the overlap.

How to approach any algebra problem

Read what is being asked: simplify an expression, or solve an equation for a letter?
Simplify each side: expand brackets and collect like terms.
Undo operations in reverse order — addition and subtraction first, then multiplication and division — doing the same to both sides.
Check by substituting your answer into the original problem.

A worked algebra problem

Solve $4(x - 2) = 2x + 6$ . Expand: $4x - 8 = 2x + 6$ . Subtract $2x$ : $2x - 8 = 6$ . Add 8: $2x = 14$ , so $x = 7$ .

Check: $4(7 - 2) = 20$ and $2(7) + 6 = 20$ . Both sides agree, so $x = 7$ is right.

Which algebra lesson do I need?

Equations with one unknown: start with equations and linear equations.
An $x^2$ in the problem: quadratic equations, then factoring polynomials.
Fractions with $x$ in them: rational expressions and rational equations.
Graphs, $f(x)$ and $y =$ : graphing equations, functions, and slope.
A story instead of symbols: how to solve word problems.

Frequently asked questions

Where can I get algebra help with my own problem?

Type it into the Sumzo algebra calculator: it shows every step, and its math engine checks the answer. The lessons here teach the methods behind those steps.

How do I check my algebra answers?

Substitute your answer into the original equation. If both sides come out equal, it is correct — if not, find the first line where your working changed the equation.

What should I learn first in algebra?

Simplifying expressions and solving one-step and two-step equations. Everything later — quadratics, functions, systems — builds on them.