Algebra & Geometry — Core Problems

★ Easy · Linear Function

What is the slope of the line passing through the points

(2, 5)

and

(6, 13)

💡 Memory: SLOPE = rise over run = (y₂ − y₁) ÷ (x₂ − x₁)

Explanation

Use the slope formula: $m = (y₂ - y₁) / (x₂ - x₁) = (13 - 5) / (6 - 2) = 8 / 4 = 2$ . Count the change in y (rise = 8) and the change in x (run = 4). Rise ÷ Run = 2.

★ Easy · Slope-Intercept Form

A line has equation

y = -3x + 7

. What is the y-intercept?

💡 Memory: In y = mx + b, the letter b IS the y-intercept.

Explanation

In slope-intercept form $y = mx + b$ , the slope is $m = -3$ and the y-intercept is $b = 7$ . The y-intercept is where the line crosses the y-axis (when x = 0). Plug in x = 0: y = −3(0) + 7 = 7.

★ Easy · Solving Equations

Solve for

x

2x + 9 = 3

💡 Memory: ISOLATE x — move constants first, then divide by the coefficient.

Explanation

Step 1: Subtract 9 from both sides → $2x = 3 - 9 = -6$ . Step 2: Divide both sides by 2 → $x = -6 / 2 = -3$ . Check: 2(−3) + 9 = −6 + 9 = 3 ✓

★★ Medium · Inequality (Tricky!)

Solve the inequality

-4x > 20

. Which answer is correct?

💡 Memory: FLIP the sign when multiplying or dividing by a NEGATIVE number!

Explanation — Most Common Mistake Here!

Divide both sides by −4. Because you divide by a NEGATIVE number, flip the inequality sign: $x < 20/(-4)$ → $x < -5$ . Most students forget to flip and choose A. Always flip when dividing/multiplying by negative!

★ Easy · Undefined Slope

Which line has an undefined slope?

💡 Memory: VERTICAL = undefined slope. HORIZONTAL = zero slope.

Explanation

x = −2 is a vertical line. In the slope formula, the run (Δx) = 0, so slope = rise/0, which is undefined (division by zero). y = 4 is horizontal with slope = 0. Think: "V for Vertical = V for undef-V-ined."

★★ Medium · Parallel Lines

Line

ℓ

has slope

3

. Which equation represents a line parallel to

ℓ

💡 Memory: PARALLEL = same slope, different y-intercept.

Explanation

Parallel lines have the same slope. The original slope is 3, so the parallel line must also have slope 3. y = 3x − 4 has slope 3 with a different y-intercept (−4), so it never intersects the original line. A has slope −1/3 (perpendicular). D has slope −3 (wrong).

★ Easy · Writing Equations

A line passes through

(0, -2)

with slope

4

. What is its equation?

💡 Memory: If you know the y-intercept, just plug into y = mx + b directly!

Explanation

The point (0, −2) is the y-intercept, so b = −2. The slope is m = 4. Substitute into y = mx + b: y = 4x − 2. Many students mix up m and b — remember m comes before x, b is the constant at the end.

★★ Medium · Solving with Variables Both Sides

Solve for

x

5x - 3 = 2x + 9

💡 Memory: COLLECT x-terms on one side first, then constants on the other.

Explanation

Step 1: Subtract 2x from both sides → $3x - 3 = 9$ . Step 2: Add 3 to both sides → $3x = 12$ . Step 3: Divide by 3 → $x = 4$ . Check: 5(4)−3 = 17 = 2(4)+9 = 17 ✓

★★ Medium · Compound Inequality (Tricky!)

Which graph represents the solution of

-1 \leq 2x + 3 < 7

💡 Memory: Do the same operation to ALL THREE parts at once.

Explanation

Subtract 3 from all three parts: $-1-3 \leq 2x < 7-3$ → $-4 \leq 2x < 4$ . Then divide all by 2: $-4/2 \leq x < 4/2$ → −2 ≤ x < 2. Key: treat it like one equation, apply operations to ALL three parts equally.

★★ Medium · Interpreting Slope (Real World)

A phone plan charges a $10 monthly fee plus $0.05 per text message. The total cost

C

for

n

texts is

C = 0.05n + 10

. What does the number 0.05 represent?

💡 Memory: SLOPE = rate of change = "per unit" change.

Explanation

In y = mx + b, the slope (m = 0.05) is the rate of change — how much the cost increases per additional text. So 0.05 = $0.05 per text. The y-intercept (b = 10) is the fixed $10 monthly fee. Slope always means "change in output per 1 unit increase in input."

★ Easy · Triangle Angle Sum

Two angles of a triangle measure

47°

and

83°

. What is the measure of the third angle?

💡 Memory: Triangle angles always add to 180°. Period.

Explanation

Sum of all three angles = 180°. Third angle = 180° − 47° − 83° = 180° − 130° = 50°. This is the Triangle Angle Sum Theorem — it works for EVERY triangle, no exceptions.

★ Easy · Area of Triangle

A triangle has a base of

10 cm

and height of

6 cm

. What is its area?

💡 Memory: Area of triangle = ½ × base × height. The ½ is the most forgotten part!

Explanation

Area = ½ × base × height = ½ × 10 × 6 = ½ × 60 = 30 cm². The most common error: forgetting the ½ and getting 60 cm². A triangle is literally half a rectangle (that's where the ½ comes from).

★★ Medium · Exterior Angle Theorem (Tricky!)

In a triangle, two interior angles measure

55°

and

70°

. What is the measure of the exterior angle at the third vertex?

💡 Memory: EXTERIOR angle = sum of the TWO non-adjacent (remote) interior angles.

Explanation

Exterior Angle Theorem: exterior angle = sum of the two remote interior angles = 55° + 70° = 125°. Alternatively: third interior angle = 180°−55°−70° = 55°, so exterior angle = 180°−55° = 125°. Choice B (115°) is a common error from subtracting instead of adding.

★ Easy · Polygon Interior Angles

What is the sum of the interior angles of a hexagon (6 sides)?

💡 Memory: Sum = (n − 2) × 180°, where n = number of sides.

Explanation

For a hexagon (n = 6): Sum = (6 − 2) × 180° = 4 × 180° = 720°. Remember the formula (n−2)×180° because a polygon can be divided into (n−2) triangles, each with 180°. Pentagon = 540°, Hexagon = 720°, Heptagon = 900°.

★ Easy · Circumference of Circle

A circle has a diameter of

10 cm

. What is its circumference? (Use

π \approx 3.14

)

💡 Memory: C = π × d (diameter) or C = 2π × r (radius). Diameter = 2 × radius!

Explanation — Watch the Units!

C = π × d = 3.14 × 10 = 31.4 cm. Notice: A says 31.4 cm² — the value is right but the unit is WRONG. Circumference is a length (cm), not area (cm²). Also, C (78.5 cm) is the area formula mistake: π × r² = 3.14 × 5² = 78.5 cm².

★★ Medium · Area of Circle

A circle has a radius of

7 cm

. What is its area? (Use

π \approx 3.14

)

💡 Memory: Area = π × r² (radius SQUARED). Not diameter, not 2r — just r²!

Explanation

Area = π × r² = 3.14 × 7² = 3.14 × 49 = 153.86 cm². Common errors: A (43.96) uses circumference formula by mistake (2πr = 2×3.14×7). D (615.44) uses diameter instead of radius (π×14² = 615.44). Always use RADIUS in the area formula!

★★ Medium · Pythagorean Theorem

A right triangle has legs of length

6

and

8

. What is the length of the hypotenuse?

💡 Memory: a² + b² = c² → c = √(a² + b²). Hypotenuse is always the LONGEST side, opposite the right angle.

Explanation

c² = 6² + 8² = 36 + 64 = 100 → c = √100 = 10. This is the famous 3-4-5 Pythagorean triple scaled by 2: (6, 8, 10) = 2×(3, 4, 5). Memorize common triples: 3-4-5, 5-12-13, 8-15-17.

★★ Medium · Inscribed Angle Theorem (Tricky!)

A central angle in a circle measures

80°

. What is the measure of an inscribed angle that intercepts the same arc?

💡 Memory: Inscribed angle = HALF the central angle (both intercept same arc).

Explanation

The Inscribed Angle Theorem: an inscribed angle is always half the central angle that intercepts the same arc. Inscribed angle = 80° ÷ 2 = 40°. Think: "Central angle is the boss — inscribed angle is always the half-sized version."

★★ Medium · Regular Polygon — Each Interior Angle

What is the measure of each interior angle of a regular octagon (8 sides)?

💡 Memory: Each angle = (n − 2) × 180° ÷ n. For regular polygons, divide total by number of angles.

Explanation

Total interior angles of octagon = (8−2)×180° = 6×180° = 1080°. Each angle in a REGULAR octagon = 1080° ÷ 8 = 135°. Note: 120° = regular hexagon, 144° = regular decagon (10 sides), 150° = regular 12-gon.

★★ Medium · Tangent Line (Tricky!)

A tangent line touches a circle at point

P

. The radius drawn to point

P

measures

5 cm

. A line segment from the center extends

13 cm

to an external point. How long is the tangent segment?

💡 Memory: TANGENT ⊥ RADIUS at point of tangency → use Pythagorean theorem!

Explanation

A tangent is perpendicular to the radius → right triangle! Hypotenuse = 13 (from center to external point), one leg = 5 (radius). tangent² = 13² − 5² = 169 − 25 = 144 → tangent = √144 = 12 cm. This is again the 5-12-13 Pythagorean triple!