Linear Equations
Solving for Variables
Isolate the variable by applying inverse operations. Keep both sides balanced.
$ax + b = c \implies x = \dfrac{c-b}{a}$
⚡ Key Rule Whatever you do to one side, do to the other.
Systems
Systems of Equations
Two methods: substitution (replace one variable) or elimination (add/subtract equations).
$\begin{cases} ax+by=c \\ dx+ey=f \end{cases}$
⚡ No Solution Parallel lines → same slope, different y-intercepts.
Inequalities
Linear Inequalities
Solve like equations but flip the inequality sign when multiplying or dividing by a negative number.
$-2x > 6 \implies x < -3$
⚡ Remember Negative ÷ or × → FLIP the sign!
Slope & Line
Slope-Intercept Form
Every non-vertical line can be expressed as $y = mx + b$ where $m$ is slope and $b$ is y-intercept.
$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\Delta y}{\Delta x}$
⚡ Parallel Same slope $m$. Perpendicular: $m_1 \cdot m_2 = -1$
📝 Worked Example — Heart of Algebra
If $3x - 7 = 2(x + 4)$, what is the value of $x$?
Show Solution ▼
Step 1: Expand the right side
$3x - 7 = 2x + 8$
Step 2: Collect x terms
$3x - 2x = 8 + 7 \implies x = 15$
✅ Answer: $x = 15$
Ratios & Proportions
Proportional Reasoning
If two quantities vary proportionally, their ratio is constant.
$\dfrac{a}{b} = \dfrac{c}{d} \implies ad = bc$
⚡ Unit Rate $\dfrac{\text{total}}{\text{units}} = \text{rate per unit}$
Percentages
Percent Change
Express change relative to the original value.
$\%\text{ change} = \dfrac{\text{new} - \text{original}}{\text{original}} \times 100$
⚡ Tip Increase by 20% → multiply by 1.20. Decrease by 15% → multiply by 0.85.
Statistics
Mean, Median, Mode
Mean = sum ÷ count. Median = middle value. Mode = most frequent.
$\bar{x} = \dfrac{\sum x_i}{n}$
⚡ Outliers Outliers affect the mean more than the median.
Probability
Basic Probability
Probability = favorable outcomes ÷ total outcomes (for equally likely events).
$P(A) = \dfrac{\text{favorable}}{\text{total}}$
⚡ Range $0 \le P(A) \le 1$. Complement: $P(A') = 1 - P(A)$
📝 Worked Example — Data Analysis
A store's revenue increased from \$4,000 to \$5,200. What is the percent increase?
Show Solution ▼
Step 1: Find the change
$5200 - 4000 = 1200$
Step 2: Divide by original
$\dfrac{1200}{4000} \times 100 = 30\%$
✅ Answer: 30% increase
Quadratics
The Quadratic Formula
For $ax^2 + bx + c = 0$, find roots using the quadratic formula.
$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
⚡ Discriminant $b^2-4ac > 0$: 2 real roots. $= 0$: 1 root. $< 0$: no real roots.
Factoring
Factoring Quadratics
Find two numbers that multiply to $ac$ and add to $b$. Factor by grouping.
$x^2 + bx + c = (x+p)(x+q)$ where $p+q=b$, $pq=c$
⚡ FOIL Check Always verify by expanding back!
Exponents
Exponent Rules
Key rules for simplifying expressions with exponents.
$a^m \cdot a^n = a^{m+n}$, $\;\dfrac{a^m}{a^n} = a^{m-n}$, $\;(a^m)^n = a^{mn}$
⚡ Zero $a^0 = 1$. Negative: $a^{-n} = \dfrac{1}{a^n}$
Functions
Function Notation & Composition
$f(x)$ means substitute $x$ into the function rule. Composition: apply one function inside another.
$(f \circ g)(x) = f(g(x))$
⚡ Order matters! $f(g(x)) \ne g(f(x))$ in general.
📝 Worked Example — Advanced Math
Solve: $x^2 - 5x + 6 = 0$
Show Solution ▼
Step 1: Find factor pairs of +6 that add to −5
$(−2) \times (−3) = 6$ and $(−2)+(−3) = −5$ ✓
Step 2: Factor
$(x-2)(x-3) = 0$
Step 3: Solve each factor
$x = 2$ or $x = 3$
✅ Answer: $x = 2$ or $x = 3$
Geometry
Essential Formulas
Area, perimeter, volume — know these cold.
Circle: $A = \pi r^2$, $C = 2\pi r$
⚡ Special Triangles 3-4-5, 5-12-13, 30-60-90, 45-45-90
Trigonometry
SOH-CAH-TOA
Right-triangle trig ratios for an angle $\theta$:
$\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$, $\;\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$, $\;\tan\theta = \dfrac{\text{opp}}{\text{adj}}$
⚡ Co-function $\sin(90°-\theta) = \cos\theta$
📝 Worked Example — Additional Topics
In a right triangle, the legs measure 5 and 12. What is the length of the hypotenuse?
Show Solution ▼
Apply Pythagorean theorem
$c^2 = 5^2 + 12^2 = 25 + 144 = 169$
Take square root
$c = \sqrt{169} = 13$
✅ Answer: Hypotenuse = 13 (5-12-13 triple!)
🎯 Start Exam Now
✏️ Review Practice
Practice — Algebra
P1. If $2(x+3) = 4x - 2$, what is $x$?
Reveal Answer ▼
$2x + 6 = 4x - 2 \implies 8 = 2x \implies x = 4$ ✅
Practice — Data
P2. A set: {3, 7, 7, 9, 14}. What is the median?
Reveal Answer ▼
Ordered: 3, 7, 7 , 9, 14 → Median = 7 ✅
Practice — Advanced
P3. Simplify: $\dfrac{x^6}{x^2}$
Reveal Answer ▼
$x^{6-2} = x^4$ ✅
Practice — Geometry
P4. What is the area of a circle with radius 6?
Reveal Answer ▼
$A = \pi r^2 = 36\pi \approx 113.1$ sq units ✅
🎯 Start Full Exam
Questions cover Heart of Algebra (5), Problem Solving & Data Analysis (6), Passport to Advanced Math (6), and Additional Topics (3). Select the best answer for each question. You'll see your score and full explanations at the end.
5 Algebra
6 Data Analysis
6 Advanced Math
3 Additional Topics
▶ Start Exam
← Prev
Q1 / 20
Next →