A high school course, on one page

Algebra makes sense once you see it balance.

Six units, from variables to graphing, each with a worked example and a problem you solve yourself. No video, no sign-up — just read, try, and check.

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2x + 3 11 x = 4
UNIT 01Variables & expressions

Foundations

Algebra just replaces unknown numbers with letters. A variable like x is a placeholder — everything else is arithmetic you already know, done in a fixed order.

Order of operations: parentheses, then exponents, then multiplication and division (left to right), then addition and subtraction (left to right).

Worked example — simplifying an expression

  • 3(x + 4) − 2xstarting expression
  • 3x + 12 − 2xdistribute the 3 across (x + 4)
  • x + 12combine like terms: 3x − 2x = x

Your turn

Simplify: 5(x − 2) + 3x
Answer in the form ax + b — enter just the coefficient of x:
5(x − 2) + 3x = 5x − 10 + 3x = 8x − 10
UNIT 02One variable, one solution

Linear Equations

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

Worked example — solving for x

  • 3(x − 2) + 4 = 13starting equation
  • 3x − 6 + 4 = 13distribute the 3
  • 3x − 2 = 13combine −6 and +4
  • 3x = 15add 2 to both sides
  • x = 5divide both sides by 3
Check your work: substitute the answer back in. 3(5 − 2) + 4 = 3(3) + 4 = 13. ✓

Your turn

Solve: 4x − 7 = 21
4x − 7 = 21 → 4x = 28 → x = 7
UNIT 03Two equations, two unknowns

Systems of Equations

A system is a pair of equations sharing the same two variables. Solving it means finding the one pair of values that satisfies both at once. Here are the two most common methods, applied to the same system.

The system: x + y = 10 and x − y = 2

Elimination — add the equations to cancel a variable

  • x + y = 10equation 1
  • x − y = 2equation 2
  • 2x = 12add the equations — the y terms cancel
  • x = 6divide by 2
  • 6 + y = 10 → y = 4substitute x = 6 into equation 1

Substitution — solve one equation for a variable, plug it into the other

  • x − y = 2 → x = y + 2solve equation 2 for x
  • (y + 2) + y = 10substitute into equation 1
  • 2y + 2 = 10combine like terms
  • y = 4solve for y
  • x = 4 + 2 = 6substitute back to find x

Your turn

Solve: x + y = 14 and x − y = 6
Adding both equations: 2x = 20 → x = 10. Then 10 + y = 14 → x = 10, y = 4
UNIT 04Two solutions, usually

Quadratic Equations

A quadratic has an x² term, which usually means two solutions instead of one. Factoring is faster when it works cleanly; the quadratic formula always works.

The equation: x² − 5x + 6 = 0

Factoring — find two numbers that multiply to 6 and add to −5

  • x² − 5x + 6 = 0starting equation
  • (x − 2)(x − 3) = 0−2 and −3 multiply to 6, add to −5
  • x − 2 = 0 or x − 3 = 0a product is zero only if a factor is zero
  • x = 2 or x = 3solve each factor

Quadratic formula — works even when factoring doesn't

Formula: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / 2a
  • a = 1, b = −5, c = 6read off the coefficients
  • x = (5 ± √(25 − 24)) / 2substitute into the formula
  • x = (5 ± 1) / 2√1 = 1
  • x = 3 or x = 2the two solutions from ±

Your turn

Solve: x² − 4 = 0  (enter the positive solution)
x² − 4 = 0 → x² = 4 → x = 2 or x = −2
UNIT 05Slope-intercept form

Functions & Graphing

A linear function y = mx + b draws a straight line. m is the slope — how steep the line is — and b is the y-intercept, where it crosses the vertical axis. Drag the sliders to see how each one changes the graph.

y = 1x + 0

Try setting m to 0 — the line goes flat, because y never changes. Try a negative m — the line slopes downward.

Your turn

A line has slope 2 and passes through (0, 3). What is its equation, y = mx + b? Enter b:
The y-intercept is where the line crosses x = 0 — that's given directly as (0, 3), so b = 3 and the equation is y = 2x + 3.
UNIT 06A range of solutions

Inequalities

Inequalities work like equations, with one crucial exception: multiplying or dividing by a negative number flips the direction of the sign.

The one rule to remember: if you multiply or divide both sides by a negative number, flip the inequality sign.

Worked example — solving and graphing

  • −2x + 3 < 7starting inequality
  • −2x < 4subtract 3 from both sides
  • x > −2divide by −2 — flip the sign
−2 −6 2

The open circle at −2 means −2 itself is not included — the solution is everything greater.

Your turn

Solve: x / 3 > 2  — enter using > or < and the number, e.g. "x>5"
x / 3 > 2 → multiply both sides by 3 (positive, so no flip) → x > 6
CHECKPOINTSix questions, mixed review

Practice

One question from each unit. Answer them all, then check your work at once.

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