Math Practice — Algebra 1 & Geometry 1

Algebra 1

Linear Equations & Expressions

10 core problems · Enter numbers only · No fractions or symbols

A-01

Solving Linear Equations

Solve for \(x\):
\[3x + 7 = 22\]

MOVE & DIVIDE — move numbers first, then divide

📖 Example

Solve \(2x + 5 = 13\)
Step 1: subtract 5 → \(2x = 8\)
Step 2: divide by 2 → \(x = 4\)

\(x\) =

A-02

Two-Step Equations · Tricky Negative

Solve for \(x\):
\[5x - 3 = 2x + 12\]

COLLECT — collect x on one side, numbers on the other

📖 Example

\(4x - 1 = x + 8\)
→ \(3x = 9\) → \(x = 3\)

\(x\) =

A-03

Distributive Property · Most Missed

Solve for \(x\):
\[2(x + 4) = 18\]

DISTRIBUTE FIRST — multiply before solving

📖 Example

\(3(x - 2) = 12\)
→ \(3x - 6 = 12\) → \(3x = 18\) → \(x = 6\)

\(x\) =

A-04

Slope of a Line · Often Confused

Find the slope of the line passing through \((2, 3)\) and \((6, 11)\).

RISE over RUN — slope = (y2-y1)/(x2-x1)

📖 Example

Points \((1,2)\) and \((3,8)\):
slope \(= \dfrac{8-2}{3-1} = \dfrac{6}{2} = 3\)

slope =

A-05

y-intercept from Slope-Intercept Form

What is the y-intercept of the line \(y = 4x - 7\)?
Enter the y-intercept value only.

y = mx + b → b IS the y-intercept

📖 Example

\(y = 3x + 5\) → y-intercept is \(5\) (where the line crosses the y-axis)

b =

A-06

Systems of Equations · Substitution

Solve the system. Find \(x\):
\[\begin{cases} y = 2x \\ x + y = 9 \end{cases}\]

PLUG IN — substitute the first equation into the second

📖 Example

\(y = x + 1,\quad x + y = 5\)
→ \(x + (x+1) = 5\) → \(2x = 4\) → \(x = 2\)

\(x\) =

A-07

Exponent Rules · Classic Trap

Simplify: \(x^3 \cdot x^4\)
What is the exponent of \(x\)?

SAME BASE ADD exponents (x^a · x^b = x^(a+b))

📖 Example

\(x^2 \cdot x^5 = x^{2+5} = x^7\) exponent = 7

exponent =

A-08

Factoring · Difference of Squares

Factor: \(x^2 - 25\)
What is the positive root? (the positive value that makes it zero)

A^2 - B^2 = (A+B)(A-B) → roots are +B and -B

📖 Example

\(x^2 - 16 = (x+4)(x-4)\) → positive root = 4

positive root =

A-09

Inequalities · Sign Flip Trap!

Solve: \(-2x > 8\)
What is \(x\)? (enter the boundary number, e.g. enter -4 if x < -4)

FLIP the sign when DIVIDING by NEGATIVE

📖 Example

\(-3x > 9\) → divide by -3, FLIP: \(x < -3\) → boundary = -3

boundary =

A-10

Quadratic · Find the Vertex x-value

For \(y = x^2 - 6x + 8\), what is the x-coordinate of the vertex?

VERTEX x = -b divided by 2a (for y = ax^2 + bx + c)

📖 Example

\(y = x^2 - 4x + 3\): a=1, b=-4 → vertex x = \(-\frac{-4}{2(1)} = 2\)

vertex x =

Geometry 1

Angles, Triangles & Circles

10 core problems · Enter numbers only · No degree symbols

G-01

Supplementary Angles

Two angles are supplementary. One angle is 65°. What is the other angle?

SUPPLEMENTARY = 180 (straight line)

📖 Example

One angle = 40° → other = 180 - 40 = 140°

angle =

G-02

Triangle Angles · Most Missed

A triangle has angles 55° and 80°. What is the third angle?

TRIANGLE SUM = 180 always

📖 Example

Angles 60° and 70° → third = 180 - 60 - 70 = 50°

third angle =

G-03

Pythagorean Theorem · Classic

A right triangle has legs 3 and 4. What is the hypotenuse?

a^2 + b^2 = c^2 → memorize 3-4-5 triangle

📖 Example

Legs 5 and 12 → \(5^2 + 12^2 = 25 + 144 = 169\) → hypotenuse = 13

hypotenuse =

G-04

Area of Triangle

A triangle has base 10 and height 6. What is the area?

HALF base times height — Area = (1/2) x b x h

📖 Example

Base = 8, height = 5 → Area = (1/2)(8)(5) = 20

area =

G-05

Vertical Angles · Tricky

Two lines intersect. One angle is 72°. What is the vertical (opposite) angle?

VERTICAL = EQUAL — opposite angles are always equal

📖 Example

Angle = 55° → vertical angle = 55° (same value, across the intersection)

vertical angle =

G-06

Circle — Circumference

A circle has diameter 10. What is the circumference?
Use π ≈ 3.14 and round to the nearest whole number.

C = pi x d (diameter) OR 2 x pi x r (radius)

📖 Example

Diameter = 5 → C = 3.14 × 5 = 15.7 → rounded ≈ 16

C ≈

G-07

Exterior Angle Theorem · Confused Often

An exterior angle of a triangle is 120°. Two interior angles are 70° and \(x\). Find \(x\).

EXTERIOR = SUM of 2 non-adjacent interior angles

📖 Example

Exterior = 100°, one interior = 60° → other = 100 - 60 = 40°

\(x\) =

G-08

Area of Circle

A circle has radius 7. What is the area?
Use π ≈ 3.14 and round to the nearest whole number.

Area = pi x r^2 (r squared, not diameter!)

📖 Example

Radius = 3 → Area = 3.14 × 9 = 28.26 ≈ 28

Area ≈

G-09

Parallel Lines — Alternate Interior Angles

Two parallel lines are cut by a transversal. One alternate interior angle is 58°. What is the other alternate interior angle?

ALTERNATE INTERIOR = EQUAL (Z-shape between parallel lines)

📖 Example

Alternate interior angle = 45° → the other alternate interior angle = 45°

angle =

G-10

Similar Triangles · Ratio Trap

Two similar triangles have corresponding sides in ratio 1:3. The smaller triangle's side is 4. What is the corresponding side of the larger triangle?

SIMILAR = PROPORTIONAL — multiply by the scale factor

📖 Example

Ratio 1:2, small side = 5 → large side = 5 × 2 = 10

large side =