Algebra lessons

Variables and constants

A variable is a letter that stands for a value we do not know yet. A constant is a fixed number that does not change.

In the expression below, x is the variable and 5 is the constant:

2x + 5

Variables can take different values. When x = 3, the expression equals 2(3) + 5 = 11.

Substitution machine

Substituting means replacing a variable with a number and calculating the result.

ax + b becomes a concrete number once you plug in a value for x.

Interactive substitution machine — slide x into ax + b.

Practice: Evaluate an expression

Evaluate a randomised expression for a given x.

Expressions, equations and identities

An expression is a combination of numbers, variables, and operations. It has no equals sign.

An equation says two expressions are equal. An identity is an equation that is true for every possible value of the variable.

Identity: (a + b)^2 = a^2 + 2ab + b^2

Equations have specific solutions. Identities hold no matter what values you substitute.

Collecting like terms

Like terms have the same variable raised to the same power. You simplify by adding or subtracting their coefficients.

Terms with different variables or powers cannot be combined. For example, 3x and 3x^2 are not like terms because the powers differ.

Simplifying step by step

Group the terms by their variable and power, then combine the coefficients.

1. \textcolor{#f871b9}{4x^2} + \textcolor{#8b5cf6}{3x} - \textcolor{#f871b9}{x^2} + \textcolor{#8b5cf6}{7x} (start)
2. (\textcolor{#f871b9}{4x^2 - x^2}) + (\textcolor{#8b5cf6}{3x + 7x}) (group like terms)
3. 3x^2 + 10x (simplified)

Watch coloured like-term chips regroup and combine.

Practice: Collect like terms

Sort terms into groups, then type the simplified result.

The distributive property

Multiplying a bracket means multiplying each term inside by the factor outside. This is called expanding.

Distributive law: a(b + c) = ab + ac

This rule also works in reverse. Pulling out a common factor is called factoring, which we cover later.

Expanding brackets

Each term inside the bracket gets multiplied by the factor outside. Follow the working below.

1. 3(x + 4)
2. 3 \cdot x + 3 \cdot 4 (distribute the 3)
3. 3x + 12 (simplified)

When two brackets are multiplied, every term in the first multiplies every term in the second. This is sometimes called FOIL (First, Outer, Inner, Last).

1. (x + 2)(x + 3)
2. \textcolor{#f871b9}{x^2} + \textcolor{#8b5cf6}{3x} + \textcolor{#3b82f6}{2x} + \textcolor{#22c55e}{6} (First, Outer, Inner, Last)
3. x^2 + 5x + 6 (collect like terms)

Animated distributive/FOIL arrows expanding brackets.

Practice: Expand brackets

Expand single or double brackets and type the result.

Solving one-step equations

Solving an equation means finding the value of the variable that makes both sides equal. The key rule: whatever you do to one side, you must do to the other.

If something is added, subtract it. If the variable is multiplied, divide. You are undoing the operation to get x by itself.

The balance scale

An equation is like a balance scale. Both sides must stay equal. When you remove something from one side, remove the same from the other.

1. 2x + \textcolor{#f871b9}{7} = 15
2. 2x = 15 - \textcolor{#f871b9}{7} = 8 (subtract 7 from both sides)
3. x = \frac{8}{\textcolor{#8b5cf6}{2}} = 4 (divide both sides by 2)

Think of it as peeling away layers. Reverse the order of operations to unwrap the variable.

Interactive balance scale: peel away terms to solve.

Practice: Solve for x

Solve a two-step linear equation for x (fractions allowed).

Inequalities

An inequality compares two expressions using <, >, \leq, or \geq instead of an equals sign.

You solve inequalities the same way as equations, with one important exception: multiplying or dividing by a negative number flips the inequality sign.

Visualising inequalities

The solution to an inequality is a range of values, not a single number. We show this on a number line.

1. \textcolor{#f871b9}{-2}x > 6
2. x \textcolor{#f871b9}{<} -3 (divide by -2, sign flips)

Solve an inequality and explore its number line.

Practice: Solve the inequality

Solve a linear inequality; see the solution on a number line.

Factoring basics

Factoring is the reverse of expanding. You rewrite an expression as a product of simpler pieces.

The first step is always to look for a common factor. Find the largest number (and variable power) that divides every term, then pull it outside a bracket.

Pulling out common factors

Think of factoring as compressing an expression back into its multiplied form.

1. 6x + 9
2. 3(2x) + 3(3) (each term has a factor of 3)
3. 3(2x + 3) (pull 3 outside)

1. 4x^2 + 8x
2. 4x(x) + 4x(2) (each term has a factor of 4x)
3. 4x(x + 2) (pull 4x outside)

Area model showing a common factor pulled out.

Factoring quadratics

A quadratic expression has the form ax^2 + bx + c. When a = 1, we need two numbers that multiply to c and add to b.

Here is why that works:

1. (x + \textcolor{#f871b9}{p})(x + \textcolor{#8b5cf6}{q}) (start with two brackets)
2. x^2 + \textcolor{#8b5cf6}{q}x + \textcolor{#f871b9}{p}x + \textcolor{#f871b9}{p}\textcolor{#8b5cf6}{q} (expand using FOIL)
3. x^2 + (\textcolor{#f871b9}{p} + \textcolor{#8b5cf6}{q})x + \textcolor{#f871b9}{p}\textcolor{#8b5cf6}{q} (collect like terms)

Comparing with x^2 + bx + c, we need:

\textcolor{#f871b9}{p} + \textcolor{#8b5cf6}{q} = b,\ \ \ \textcolor{#f871b9}{p}\textcolor{#8b5cf6}{q} = c

The diamond method

A quick way to find the two numbers: draw a diamond (X shape). Put the product at the top and the sum at the bottom. The two numbers go on the sides.

1. x^2 + 5x + 6 (factor this)
2. p \cdot q = 6,\ \ p + q = 5 (find p and q)
3. p = 2,\ \ q = 3 (2 times 3 = 6, 2 + 3 = 5)
4. (x + 2)(x + 3) (factored form)

Interactive diamond method for factoring quadratics.

Practice: Factor the quadratic

Factor a quadratic into two brackets.

The quadratic formula

Not every quadratic factors neatly. The quadratic formula finds the roots of any equation ax^2 + bx + c = 0:

Quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The part under the square root, b^2 - 4ac, is called the discriminant.

It tells you how many real solutions exist:

1. b^2 - 4ac > 0 (two solutions)
2. b^2 - 4ac = 0 (one solution)
3. b^2 - 4ac < 0 (no real solutions)

The discriminant in action

The discriminant controls whether the parabola crosses the x-axis, touches it, or stays above/below it.

Slide a, b, c and watch the discriminant and roots.

Practice: Quadratic formula

Use the quadratic formula to find both roots.

Simultaneous equations

Sometimes you have two unknowns and two equations. You need both equations together to find a unique solution.

The elimination method adds or subtracts the equations to cancel one variable. The substitution method solves one equation for a variable and plugs it into the other.

Where two lines meet

Two linear equations represent two straight lines. The solution is the point where they cross.

1. 2x + y = 10 (equation 1)
2. x - y = 2 (equation 2)
3. 3x = 12 (add both equations)
4. x = 4 (divide by 3)
5. y = 10 - 2(4) = 2 (substitute back)

Two lines meeting — drag one and watch the crossing point.

Practice: Simultaneous equations

Solve a pair of simultaneous equations for x and y.

Exponent rules

Exponents follow consistent rules that make simplifying much faster. These three cover most situations:

1. a^m \times a^n = a^{m+n} (product rule)
2. \frac{a^m}{a^n} = a^{m-n} (quotient rule)
3. (a^m)^n = a^{mn} (power rule)

Two special cases to remember:

a^0 = 1

A negative exponent means a reciprocal: a^{-n} = \frac{1}{a^n}.

Exponent builder

See how the rules work by building expressions and watching the exponents combine.

Exponent builder — combine powers with the three rules.

Practice: Simplify the expression

Apply an exponent rule and type the simplified power.

Rearranging formulas

Making a variable the subject of a formula uses the same balancing rules as solving equations. The goal is to isolate the target variable on one side.

Treat every other letter as if it were a number.

Rearranging in action

Watch the formula transform as we isolate t:

1. v = \textcolor{#8b5cf6}{u} + \textcolor{#f871b9}{a}t (original formula)
2. v - \textcolor{#8b5cf6}{u} = \textcolor{#f871b9}{a}t (subtract u from both sides)
3. \frac{v - \textcolor{#8b5cf6}{u}}{\textcolor{#f871b9}{a}} = t (divide both sides by a)

The same process works for any formula. Subtract, divide, or take roots until the target variable stands alone.

Tap a variable to make it the subject, step by step.

Practice: Rearrange the formula

Rearrange a formula for the named variable.