⚡ Memory Key
RISE over RUN = (y₂−y₁) / (x₂−x₁)
Find the slope of the line passing through
\((2, 5)\) and \((6, 13)\).
★ Tricky: don't flip the order of subtraction!
📘 Step-by-Step
1Label the points: \((x_1,y_1)=(2,5)\) and \((x_2,y_2)=(6,13)\)
2Apply the formula: \(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{13-5}{6-2} = \dfrac{8}{4} = 2\)
✅ The slope is 2. Common mistake: subtracting in different orders (e.g., \(\frac{5-13}{6-2}\) gives −2).
⚡ Memory Key
y = mx + b · m = slope · b = y-intercept
Which equation has slope \(-3\) and y-intercept \(7\)?
★ Watch the sign of the slope carefully.
📘 Explanation
1In \(y = mx + b\), the coefficient of \(x\) is the slope \(m\).
2Slope \(= -3\) and y-intercept \(= 7\) → \(y = -3x + 7\).
⚠️ Option D swaps slope and intercept — a very common error!
⚡ Memory Key
y − y₁ = m(x − x₁) · plug IN the point
Write the equation of a line with slope \(4\) passing through \((1, -2)\).
★ Tricky sign: notice the y-value is negative.
📘 Step-by-Step
1Use point-slope: \(y - (-2) = 4(x - 1)\)
2Simplify: \(y + 2 = 4x - 4\)
3Subtract 2: \(y = 4x - 6\)
⚠️ Option D looks like point-slope but uses \(-(-2) = +2\) incorrectly.
⚡ Memory Key
VERTICAL = x = # · HORIZONTAL = y = # · "V comes before H"
What is the equation of a vertical line through \((−4, 3)\)?
★ Most-missed: students confuse x = # with y = #.
📘 Explanation
1A vertical line is always written as \(x = \text{constant}\).
2The x-coordinate of the point is \(-4\), so: \(x = -4\).
🔑 Vertical lines have undefined slope. Horizontal lines have slope = 0.
⚡ Memory Key
PARALLEL = SAME slope · PERPENDICULAR = NEGATIVE RECIPROCAL
Which line is parallel to \(y = \dfrac{2}{3}x - 5\)?
★ Parallel means equal slopes but different y-intercepts.
📘 Explanation
1Parallel lines share the same slope. Original slope = \(\dfrac{2}{3}\).
2Option B has slope \(\dfrac{2}{3}\) with different intercept → ✅ Parallel.
⚠️ Option A (\(-\frac{3}{2}\)) is the perpendicular slope (negative reciprocal).
⚡ Memory Key
x-intercept: SET y = 0 · y-intercept: SET x = 0
Find the x-intercept of \(3x - 6y = 12\).
★ Don't confuse which variable to set to zero!
📘 Step-by-Step
1Set \(y = 0\): \(3x - 6(0) = 12\)
2Solve: \(3x = 12 \Rightarrow x = 4\)
✅ x-intercept = \((4, 0)\). The x-intercept always has \(y = 0\)!
⚡ Memory Key
Ax + By = C → slope = −A/B
What is the slope of \(4x + 2y = 10\)?
★ Solve for y first — don't skip the division step!
📘 Step-by-Step
1Isolate \(y\): \(2y = -4x + 10\)
2Divide by 2: \(y = -2x + 5\)
✅ Slope = \(-2\). Common mistake: forgetting to divide the whole right side by 2.
⚡ Memory Key
Count RISE (↑↓) then RUN (→←) between two clear grid points
A line passes through \((0, 4)\) and \((3, 0)\). What is its slope?
★ Going down = negative rise. Don't forget the sign!
📘 Explanation
1\(m = \dfrac{0-4}{3-0} = \dfrac{-4}{3}\)
✅ The line goes down as x increases → negative slope. Rise = −4, Run = 3.
⚡ Memory Key
slope = RATE of change · y-intercept = STARTING value
A taxi charges \$3 base fare + \$2 per mile. Which function models the total cost \(C\) for \(m\) miles?
📘 Explanation
1Rate = $2/mile → slope = 2
2Starting cost (base fare) = $3 → y-intercept = 3
✅ \(C = 2m + 3\). Students often swap the slope and intercept values.
⚡ Memory Key
PERPENDICULAR: flip & flip sign → m × m⊥ = −1
A line has equation \(y = \dfrac{1}{3}x + 2\). What is the slope of a perpendicular line?
★ Very commonly missed: students forget to flip AND change sign.
📘 Explanation
1Original slope: \(m = \dfrac{1}{3}\)
2Perpendicular slope: flip → \(3\), then change sign → \(-3\)
✅ Check: \(\dfrac{1}{3} \times (-3) = -1\) ✔
⚡ Memory Key
Triangle angles ALWAYS add to 180°
A triangle has angles \(52°\) and \(79°\). What is the third angle?
📘 Explanation
✅ Third angle = \(49°\).
⚡ Memory Key
a² + b² = c² · c = HYPOTENUSE (longest side, opposite right angle)
A right triangle has legs \(a = 6\) and \(b = 8\). What is the hypotenuse?
★ Don't just add the legs! You must square, add, then square root.
📘 Step-by-Step
1\(c^2 = 6^2 + 8^2 = 36 + 64 = 100\)
✅ This is the classic 3-4-5 triple scaled by 2: (6, 8, 10).
⚡ Memory Key
SUPPLEMENTARY = 180° · COMPLEMENTARY = 90° · "S is Straight (line)"
If two angles are supplementary and one measures \(112°\), what is the other?
📘 Explanation
1Supplementary → sum = 180°
⚡ Memory Key
Area = πr² · Circumference = 2πr · "Pie Are Squared"
A circle has radius \(5\). What is its area? (Leave in terms of \(\pi\).)
★ Don't confuse the area formula with the circumference formula!
📘 Explanation
1Area \(= \pi r^2 = \pi \cdot 5^2 = 25\pi\)
⚠️ \(10\pi\) is the circumference \((2\pi \cdot 5)\). Don't mix them up!
⚡ Memory Key
VERTICAL angles are EQUAL · they're across from each other (X shape)
Two lines intersect forming vertical angles. One angle measures \((3x + 10)°\) and its vertical angle measures \((5x - 8)°\). Find \(x\).
📘 Step-by-Step
1Vertical angles are equal: \(3x + 10 = 5x - 8\)
2\(18 = 2x \Rightarrow x = 9\)
✅ Check: \(3(9)+10 = 37°\) and \(5(9)-8 = 37°\) ✔
⚡ Memory Key
EXTERIOR angle = SUM of the two NON-adjacent interior angles
In a triangle, two interior angles are \(40°\) and \(65°\). What is the exterior angle at the third vertex?
★ Many students subtract from 180° first. Skip that step!
📘 Explanation
1Exterior angle = sum of two remote interior angles
✅ You can verify: third interior angle = \(180-105 = 75°\). \(40+65+75 = 180°\) ✔
⚡ Memory Key
ALTERNATE INTERIOR = equal · CO-INTERIOR (same-side) = 180°
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is \(72°\). Find the other co-interior angle.
★ Co-interior angles are supplementary — they add to 180°, not equal!
📘 Explanation
1Co-interior (consecutive interior) angles are supplementary: sum = 180°
⚠️ Alternate interior angles would be equal (72°) — don't confuse the two types!
⚡ Memory Key
COMPOSITE = SPLIT into simple shapes · Add or Subtract areas
An L-shaped figure is formed by a \(6 \times 8\) rectangle with a \(2 \times 3\) rectangle cut from one corner. What is the area?
📘 Step-by-Step
1Large rectangle: \(6 \times 8 = 48\)
2Removed piece: \(2 \times 3 = 6\)
⚡ Memory Key
SIMILAR = same shape, different size · RATIOS of sides are EQUAL
Triangle A has sides 3, 4, 5. Triangle B is similar with shortest side 9. What is the longest side of Triangle B?
📘 Step-by-Step
1Scale factor: \(\dfrac{9}{3} = 3\)
2Longest side of A is 5 → \(5 \times 3 = 15\)
✅ All sides scale by the same factor: 3→9, 4→12, 5→15.
⚡ Memory Key
V = πr²h · "Circle bottom × height" · r is RADIUS not diameter!
A cylinder has diameter \(6\) cm and height \(10\) cm. What is its volume? (Leave in terms of \(\pi\).)
★ Super common mistake: using diameter 6 instead of radius 3!
📘 Step-by-Step
1Radius = diameter ÷ 2 = \(6 ÷ 2 = 3\) cm
2\(V = \pi r^2 h = \pi \cdot 3^2 \cdot 10 = \pi \cdot 9 \cdot 10 = 90\pi\)
⚠️ \(360\pi\) comes from using diameter 6 as radius → classic trap!