Algebra 1
10 core problems · Linear equations, inequalities, systems, functions, quadratics, and exponents.
01
ISOLATE → COMBINE → DIVIDE — always move the variable to one side first, then constants to the other.
Worked Example
Solve: \(3x + 7 = 22\)
Step 1: subtract 7 → \(3x = 15\)
Step 2: divide by 3 → \(x = 5\)
Step 1: subtract 7 → \(3x = 15\)
Step 2: divide by 3 → \(x = 5\)
Solve for \(x\):
\[5x - 3 = 2x + 12\] Hint: move the \(x\) terms to one side first.
\[5x - 3 = 2x + 12\] Hint: move the \(x\) terms to one side first.
x =
02
FLIP the sign when you multiply or divide by a NEGATIVE number. Easiest thing to forget!
Worked Example
Solve: \(-2x > 6\)
Divide by −2, flip sign → \(x < -3\)
Divide by −2, flip sign → \(x < -3\)
Solve for \(x\):
\[-4x + 1 \geq 13\] Write your answer as: x ≤ or x ≥ (number), e.g. x≤-3
\[-4x + 1 \geq 13\] Write your answer as: x ≤ or x ≥ (number), e.g. x≤-3
Answer:
03
y = mx + b · m = slope (rise/run) · b = y-intercept · Parallel lines: same slope. Perpendicular: NEGATIVE RECIPROCAL.
Worked Example
Line through (0, 3) with slope 2 → \(y = 2x + 3\)
A line passes through the points \((2,\, 1)\) and \((6,\, 9)\).
What is the slope of the line?
What is the slope of the line?
m =
04
SUBSTITUTE → SOLVE → BACK-SUBSTITUTE · Solve one equation for a variable, plug into the other.
Worked Example
\(y = x + 2\) and \(2x + y = 8\)
Sub \(y\): \(2x + (x+2) = 8\) → \(3x = 6\) → \(x = 2,\; y = 4\)
Sub \(y\): \(2x + (x+2) = 8\) → \(3x = 6\) → \(x = 2,\; y = 4\)
Solve the system:
\[\begin{cases} y = 2x - 1 \\ 3x + y = 14 \end{cases}\] What is the value of \(x\)?
\[\begin{cases} y = 2x - 1 \\ 3x + y = 14 \end{cases}\] What is the value of \(x\)?
x =
05
ADD exponents when MULTIPLYING · SUBTRACT when DIVIDING · MULTIPLY when raising a POWER to a POWER
Worked Example
\(x^3 \cdot x^4 = x^{3+4} = x^7\)
\((x^2)^3 = x^{2 \times 3} = x^6\)
\((x^2)^3 = x^{2 \times 3} = x^6\)
Simplify:
\[\frac{x^8}{x^3}\] Write your answer as \(x^n\) (just type the exponent, e.g. 5).
\[\frac{x^8}{x^3}\] Write your answer as \(x^n\) (just type the exponent, e.g. 5).
x^
06
For \(x^2 + bx + c\): find two numbers that MULTIPLY to c and ADD to b. Call them \(p, q\) → \((x+p)(x+q)\)
Worked Example
Factor \(x^2 + 5x + 6\)
Find: multiply to 6, add to 5 → 2 and 3
Answer: \((x+2)(x+3)\)
Find: multiply to 6, add to 5 → 2 and 3
Answer: \((x+2)(x+3)\)
Factor completely:
\[x^2 - 7x + 10\] Write answer as (x−a)(x−b), type just the two numbers separated by comma, e.g. 2,5
\[x^2 - 7x + 10\] Write answer as (x−a)(x−b), type just the two numbers separated by comma, e.g. 2,5
(x−a)(x−b) where a,b =
07
x = (−b ± √(b²−4ac)) / 2a · The discriminant b²−4ac: positive → 2 roots, zero → 1 root, negative → no real roots
Worked Example
Solve \(x^2 - 5x + 6 = 0\): \(a=1, b=-5, c=6\)
\(x = \frac{5 \pm \sqrt{25-24}}{2} = \frac{5 \pm 1}{2}\) → \(x = 3\) or \(x = 2\)
\(x = \frac{5 \pm \sqrt{25-24}}{2} = \frac{5 \pm 1}{2}\) → \(x = 3\) or \(x = 2\)
How many real solutions does the equation have?
\[2x^2 + 3x + 5 = 0\] Just type: 0, 1, or 2
\[2x^2 + 3x + 5 = 0\] Just type: 0, 1, or 2
Number of solutions:
08
f(x) means "plug x in." DOMAIN = all allowed inputs. For square roots: inside ≥ 0. For fractions: denominator ≠ 0.
Worked Example
If \(f(x) = 3x - 2\), then \(f(4) = 3(4) - 2 = 10\)
Given \(f(x) = x^2 - 3x + 1\),
find \(f(4)\).
find \(f(4)\).
f(4) =
09
% change = (new − old) / old × 100 · Positive = increase, negative = decrease
Worked Example
Price goes from $40 to $50: \(\frac{50-40}{40} \times 100 = 25\%\) increase
A store originally priced a jacket at $80. It is now on sale for $60.
What is the percent decrease? (Enter a number, no % symbol)
What is the percent decrease? (Enter a number, no % symbol)
% decrease =
10
READ → DEFINE → TRANSLATE → SOLVE · Let a variable = the unknown, write the equation from the problem's conditions.
Worked Example
"Three times a number decreased by 4 equals 14." → \(3n - 4 = 14\) → \(n = 6\)
Maria and Tom together have 35 books. Maria has 5 more books than Tom.
How many books does Tom have?
How many books does Tom have?
Tom =
Geometry 1
10 core problems · Angles, triangles, area, the Pythagorean theorem, circles, and congruence.
01
Complementary = Corner (90°) · Supplementary = Straight line (180°)
Worked Example
Supplement of 65°: \(180° - 65° = 115°\)
Two angles are supplementary. One angle measures \(3x + 10\)° and the other measures \(x + 30\)°.
Find \(x\).
Find \(x\).
x =
02
ALL triangles: angles add to 180° · Exterior angle = sum of the two NON-ADJACENT interior angles
Worked Example
Angles 50° and 70° → third angle = \(180 - 50 - 70 = 60°\)
A triangle has angles \(2x°\), \(3x°\), and \(4x°\).
What is the value of \(x\)?
What is the value of \(x\)?
x =
03
a² + b² = c² · c is ALWAYS the hypotenuse (longest side, opposite the right angle). Common triples: 3-4-5, 5-12-13, 8-15-17
Worked Example
Legs 6 and 8: \(6^2 + 8^2 = 36 + 64 = 100\) → \(c = \sqrt{100} = 10\)
A right triangle has legs of length 9 and 12.
What is the length of the hypotenuse?
What is the length of the hypotenuse?
c =
04
Rectangle: A = l×w · Triangle: A = ½bh · Trapezoid: A = ½(b₁+b₂)h · Circle: A = πr²
Worked Example
Triangle with base 10, height 6: \(A = \frac{1}{2}(10)(6) = 30\)
A trapezoid has parallel bases of 8 cm and 14 cm, and a height of 5 cm.
What is the area? (in cm²)
What is the area? (in cm²)
Area =
05
C = 2πr = πd (circumference) · A = πr² (area) · Always check: is the given number the RADIUS or DIAMETER?
Worked Example
Circle with diameter 10: radius = 5 → \(C = 2\pi(5) = 10\pi\)
A circle has a diameter of \(12\).
What is the area? Express as a multiple of \(\pi\), e.g. type: 36 (for 36π)
What is the area? Express as a multiple of \(\pi\), e.g. type: 36 (for 36π)
Area =
π
06
SSS · SAS · ASA · AAS · HL (right triangles) — these prove congruence. SSA and AAA do NOT prove congruence!
Worked Example
Two triangles share a side. Two pairs of sides are equal → SSS if three sides match, SAS if the equal angle is between the two equal sides.
Two triangles have: two pairs of equal sides, and the included angle (between those sides) is equal in both triangles.
Which congruence theorem applies? (Type the abbreviation, e.g. SSS)
Which congruence theorem applies? (Type the abbreviation, e.g. SSS)
Theorem:
07
Parallel lines + transversal: Alternate Interior = EQUAL · Corresponding = EQUAL · Co-interior (same-side) = 180°
Worked Example
Parallel lines cut by transversal. Angle = 70° → alternate interior angle also = 70°.
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is \((2x + 15)°\) and the other is \((3x + 10)°\).
Find \(x\).
Find \(x\).
x =
08
Similar = same SHAPE, different SIZE · Corresponding sides are PROPORTIONAL · Set up: a/b = c/d → cross multiply
Worked Example
Triangles similar. Side 4 corresponds to 10; side 6 corresponds to \(x\).
\(\frac{4}{10} = \frac{6}{x}\) → \(x = 15\)
\(\frac{4}{10} = \frac{6}{x}\) → \(x = 15\)
Two similar triangles have corresponding sides. In the smaller triangle, a side is 5. The corresponding side in the larger triangle is 20. Another side of the smaller triangle is 9.
What is the corresponding side in the larger triangle?
What is the corresponding side in the larger triangle?
Side =
09
Prism/Cylinder: V = Base Area × height · Pyramid/Cone: V = ⅓ × Base Area × height · Sphere: V = (4/3)πr³
Worked Example
Rectangular prism: \(l=4, w=3, h=5\) → \(V = 4 \times 3 \times 5 = 60\)
A cone has a base radius of 3 and a height of 7.
What is the volume? Express as a multiple of \(\pi\), e.g. type 21 for 21π.
What is the volume? Express as a multiple of \(\pi\), e.g. type 21 for 21π.
V =
π
10
d = √((x₂−x₁)² + (y₂−y₁)²) — it's the Pythagorean theorem in disguise! Horizontal diff = leg, vertical diff = leg, distance = hypotenuse.
Worked Example
Distance from (1,1) to (4,5):
\(d = \sqrt{(4-1)^2 + (5-1)^2} = \sqrt{9+16} = \sqrt{25} = 5\)
\(d = \sqrt{(4-1)^2 + (5-1)^2} = \sqrt{9+16} = \sqrt{25} = 5\)
Find the distance between points \(A(0,\, 0)\) and \(B(6,\, 8)\).
d =