Part 1 — Algebra I
Linear Functions · Equations · Inequalities
Which of the following is the slope of the line passing through $(2, 5)$ and $(6, 13)$?
Remember: slope $= \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$
⚡ Quick Memory
SLOPE = RISE ÷ RUN → subtract y's on top, x's on bottom. Order must match!
📖 Explanation
Slope $m = \dfrac{13 - 5}{6 - 2} = \dfrac{8}{4} = \mathbf{2}$.✦ Common mistake: students subtract in the wrong order (e.g., $\frac{5-13}{6-2}$). Always keep the same order: second point minus first point for both $y$ and $x$.
The equation of a line is $y = -3x + 7$. What is the $y$-intercept?
Form: $y = mx + b$ where $b$ is the $y$-intercept.
⚡ Quick Memory
y = mx + b → b = where the line Begins (crosses y-axis)
📖 Explanation
In $y = mx + b$, the coefficient of $x$ is the slope ($m = -3$) and the constant is the $y$-intercept ($b = 7$).✦ Trap: Many students confuse $m$ and $b$. Remember $b$ stands alone — no $x$ attached!
Solve for $x$: $\quad 2x + 9 = 25$
Isolate $x$ by doing the same operation to both sides.
⚡ Quick Memory
INVERSE OPERATIONS: subtract before dividing. Undo in reverse order!
📖 Explanation
$2x + 9 = 25$ → subtract 9: $2x = 16$ → divide by 2: $x = \mathbf{8}$.✦ Trap: Dividing before subtracting gives $x = 17$ — wrong! Always subtract the constant first.
Which value of $x$ satisfies $\dfrac{x-3}{2} = \dfrac{x+1}{4}$?
Cross-multiply or multiply both sides by the LCD.
⚡ Quick Memory
CROSS-MULTIPLY: $\dfrac{a}{b} = \dfrac{c}{d}$ → $a \cdot d = b \cdot c$
📖 Explanation
Multiply both sides by 4: $2(x-3) = x+1$ → $2x - 6 = x + 1$ → $x = \mathbf{7}$.✦ Common error: forgetting to distribute the 2 → writing $2x - 3$ instead of $2x - 6$.
Solve: $-4x + 3 > 19$. Which number line shows the correct solution?
Remember the key rule when multiplying or dividing by a negative number!
⚡ Quick Memory
FLIP THE SIGN when multiplying or dividing both sides by a NEGATIVE!
📖 Explanation
$-4x + 3 > 19$ → $-4x > 16$ → divide by $-4$ and flip: $x < -4$.✦ Most common error: forgetting to flip the inequality sign when dividing by $-4$.
Line $\ell$ has slope $\frac{2}{3}$ and passes through $(0, -1)$. What is $y$ when $x = 6$?
Write the equation first using slope-intercept form, then substitute.
⚡ Quick Memory
PLUG-IN: Write equation → substitute x → calculate y. One step at a time!
📖 Explanation
Equation: $y = \frac{2}{3}x - 1$. At $x = 6$: $y = \frac{2}{3}(6) - 1 = 4 - 1 = \mathbf{3}$.✦ Trap: Adding instead of subtracting the $y$-intercept: $4 + 1 = 5$ — wrong!
Which point is the solution to the system?
$$\begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases}$$
$$\begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases}$$
Set the two expressions for $y$ equal to each other.
⚡ Quick Memory
SUBSTITUTION: if both = y, set them EQUAL. Solve for x, then find y.
📖 Explanation
$2x - 1 = -x + 5$ → $3x = 6$ → $x = 2$. Then $y = 2(2)-1 = \mathbf{3}$. Answer: $(2, 3)$.✦ Always verify by plugging into both equations: $3 = -2+5 = 3$ ✓
Which compound inequality represents: "all numbers between $-2$ and $5$, including $5$ but not $-2$"?
Open circle = not included; closed circle = included.
⚡ Quick Memory
OPEN bracket "(" / strict < > → NOT included. CLOSED bracket "[" / ≤ ≥ → INCLUDED.
📖 Explanation
"$-2$ not included" → strict $<$ on the left. "$5$ included" → $\le$ on the right. So: $-2 < x \le 5$.✦ Read carefully: "including 5 but not −2" — one side is strict, one side is not!
Line $A$ has equation $y = 4x - 2$. Line $B$ is perpendicular to $A$ and passes through $(8, 1)$. What is the equation of Line $B$?
Perpendicular lines have slopes that are negative reciprocals.
⚡ Quick Memory
PERPENDICULAR slope = FLIP and FLIP SIGN. (e.g. 4 → -¼)
📖 Explanation
Perpendicular slope: $m_\perp = -\frac{1}{4}$.Point-slope: $y - 1 = -\frac{1}{4}(x - 8)$ → $y = -\frac{1}{4}x + 2 + 1 = \mathbf{-\frac{1}{4}x + 3}$.
✦ Trap A: forgetting to flip the sign. Trap B: using the original slope.
Maya earns $\$12$ per hour. She has already saved $\$45$ and wants to save at least $\$200$ total. What is the minimum number of whole hours she must work?
Set up an inequality, then round up if the answer is not a whole number.
⚡ Quick Memory
"AT LEAST" = ≥ (greater than or equal). Always ROUND UP for minimum whole hours!
📖 Explanation
$12h + 45 \ge 200$ → $12h \ge 155$ → $h \ge 12.916\ldots$Since $h$ must be a whole number: $h = \mathbf{13}$ hours.
✦ Trap: Choosing 12 hours (not enough: $12 \times 12 + 45 = 189 < 200$).
Part 2 — Geometry
Triangles · Polygons · Circles
In a triangle, two angles measure $55°$ and $72°$. What is the third angle?
The sum of angles in any triangle is always $180°$.
⚡ Quick Memory
TRIANGLE ANGLES always add to 180°. Missing angle = 180 − (sum of other two)
📖 Explanation
$180° - 55° - 72° = \mathbf{53°}$.✦ Quick check: $55 + 72 + 53 = 180$ ✓. Always verify your answer adds to 180!
A right triangle has legs of length $6$ and $8$. What is the length of the hypotenuse?
Use the Pythagorean theorem: $a^2 + b^2 = c^2$
⚡ Quick Memory
PYTHAGOREAN: a² + b² = c² → c = hypotenuse (longest side, opposite right angle)
📖 Explanation
$6^2 + 8^2 = 36 + 64 = 100 = c^2$ → $c = \sqrt{100} = \mathbf{10}$.✦ This is the famous 3-4-5 triple doubled: $6 = 2\times3,\, 8 = 2\times4,\, 10 = 2\times5$.
What is the sum of the interior angles of a regular hexagon?
Formula for sum of interior angles: $(n - 2) \times 180°$, where $n$ = number of sides.
⚡ Quick Memory
INTERIOR ANGLE SUM = (n − 2) × 180. Subtract 2 sides (always!), then multiply by 180.
📖 Explanation
Hexagon: $n = 6$. Sum $= (6-2)\times180° = 4\times180° = \mathbf{720°}$.✦ Trap: Using $(n)\times180°$ instead of $(n-2)\times180°$ gives $1080°$ — wrong!
A circle has a radius of $7$ cm. What is its circumference? (Use $\pi \approx 3.14$)
Circumference $C = 2\pi r$
⚡ Quick Memory
C = 2πr (circumference uses RADIUS) — or C = πd (diameter). "Cherry PIE" = C = πd!
📖 Explanation
$C = 2\pi r = 2 \times 3.14 \times 7 = \mathbf{43.96}$ cm.✦ Trap A: Using $\pi r$ only (forgot the 2). Trap D: That's the area formula $\pi r^2$!
A circle has an area of $100\pi$ cm². What is its radius?
Area formula: $A = \pi r^2$. Solve for $r$.
⚡ Quick Memory
Area = π r² → divide both sides by π, then SQUARE ROOT to get r.
📖 Explanation
$\pi r^2 = 100\pi$ → $r^2 = 100$ → $r = \sqrt{100} = \mathbf{10}$ cm.✦ Trap: $r = 100$ (forgot to square root). Trap B: $r = 50$ (divided by 2 instead of square-rooting).
Two triangles have two sides and the included angle equal. Which congruence postulate applies?
The key word is "included" — the angle is between the two known sides.
⚡ Quick Memory
SAS = Side-ANGLE-Side (angle BETWEEN sides). SSA ≠ congruence (not a valid postulate!)
📖 Explanation
Two sides + the angle between them = SAS (Side-Angle-Side).✦ If the angle were NOT between the sides, SSA would apply — but SSA is NOT a valid congruence postulate!
Each exterior angle of a regular polygon measures $40°$. How many sides does the polygon have?
The sum of all exterior angles of any convex polygon is always $360°$.
⚡ Quick Memory
SUM of exterior angles = always 360°. Number of sides = 360 ÷ each exterior angle.
📖 Explanation
$n = \dfrac{360°}{40°} = \mathbf{9}$ sides (a nonagon).✦ Trap: Using the interior angle formula — the exterior angle is what you directly divide 360 by.
Triangle $ABC \sim$ Triangle $DEF$. If $AB = 6$, $DE = 9$, and $BC = 8$, what is $EF$?
Similar triangles have proportional sides. Set up a proportion.
⚡ Quick Memory
SIMILAR: corresponding sides are PROPORTIONAL. AB/DE = BC/EF. Cross-multiply!
📖 Explanation
$\dfrac{AB}{DE} = \dfrac{BC}{EF}$ → $\dfrac{6}{9} = \dfrac{8}{EF}$ → $EF = \dfrac{8 \times 9}{6} = \mathbf{12}$.✦ Trap: Matching sides in wrong order. Always match the same position in each triangle!
A circle has radius $10$ cm. A sector has a central angle of $72°$. What is the arc length of the sector?
Arc length $= \dfrac{\theta}{360°} \times 2\pi r$
⚡ Quick Memory
ARC LENGTH = (angle/360) × circumference. Think: "fraction of the full circle."
📖 Explanation
Arc $= \dfrac{72}{360} \times 2\pi(10) = \dfrac{1}{5} \times 20\pi = \mathbf{4\pi}$ cm.✦ Trap: Using the area formula ($\pi r^2$) instead of circumference ($2\pi r$) for arc length.
A regular pentagon has a side length of $8$ cm and an apothem of $5.5$ cm. What is its area?
Area of a regular polygon $= \dfrac{1}{2} \times \text{perimeter} \times \text{apothem}$
⚡ Quick Memory
POLYGON AREA = ½ × Perimeter × Apothem. Apothem = distance from center to side (NOT vertex).
📖 Explanation
Perimeter $= 5 \times 8 = 40$ cm.Area $= \frac{1}{2} \times 40 \times 5.5 = \mathbf{110}$ cm².
✦ Trap D: Forgetting the $\frac{1}{2}$ gives $220$ — always halve it!