Algebra I: The Grammar of the Unknown

A variable is a name for a quantity whose value we have not pinned down yet.

An expression is a sum of terms, and only like terms — those sharing the exact same variable part — may be combined by adding their coefficients.

The order of operations is a shared convention that makes one written expression yield exactly one value for every reader.

Translating a situation into algebra means defining the variable precisely, converting the words operation by operation, and reading the result back as the original sentence.

An equation asserts that two expressions name the same number, so any operation applied identically to both sides preserves that truth.

A two-step equation is solved by undoing its operations in reverse order: the last operation applied is the first one undone.

When the variable appears on both sides, simplify each side, collect the variable terms with one balance move, and let the surviving sentence decide the case.

A formula is an equation with several letters; the same balance moves isolate whichever letter you need, treating the others as numbers you have not been told.

An ordered pair (x, y) gives every point on the plane a unique address, measured from the origin along two perpendicular number lines.

A relation is a function exactly when each input produces one and only one output; f(x) names that single output.

Slope is the constant rate at which a line trades rise for run — the same number between any two of its points.

In y = mx + b, the intercept b and the slope m carry the whole line — enough to graph it at sight, or to write it from two points.

A system's solution is the one pair (x, y) that makes both equations true at once — the point where the two lines cross.

Substitution and elimination find a system's exact solution: replace a variable with the expression it equals, or add scaled equations so a variable cancels — then back-substitute and check in both originals.

An exponent counts repeated factors, and every exponent law — including a⁰ = 1 and a⁻ⁿ = 1/aⁿ — follows from counting the factors or continuing the dividing pattern.

A product equals zero only when a factor does, so factoring x² + bx + c into (x + p)(x + q) — product c, sum b — turns one quadratic equation into two one-step equations.