Math topic explainers

A linear equation states that two expressions are equal. Solving it means isolating the variable using operations that keep both sides balanced: add, subtract, multiply or divide the same quantity on each side.

Slope measures how steeply a line rises or falls: the change in y divided by the change in x between any two points on the line, m = (y₂ − y₁)/(x₂ − x₁).

A system asks for the values that satisfy two or more equations at once. Graphically, that is the point where the lines cross. Algebraically, you can substitute one equation into the other or eliminate a variable by adding.

Factoring rewrites a quadratic as a product of two binomials. For x² + bx + c, look for two numbers that multiply to c and add to b. When a ≠ 1, split the middle term using two numbers that multiply to ac.

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). The discriminant b² − 4ac tells you how many real solutions exist before you compute them.

A function assigns exactly one output to each input. f(x) is read 'f of x' and names the output for input x — it is not multiplication. Domain is the set of allowed inputs; range is the set of resulting outputs.

In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. Choosing the ratio is a matter of labelling the sides relative to the angle you are given.

Probability measures how likely an outcome is, on a scale from 0 to 1. Multiply probabilities for events that must both happen; add them for mutually exclusive alternatives, subtracting the overlap when there is one.

A limit asks what value a function approaches as x approaches a point — not what happens at the point. When direct substitution gives 0/0, the expression usually simplifies first.