DSAT Math Concepts and Techniques

Linear Equations in One Variable

Form: ax + b = c

Step 1 (Linear Equations)

Subtract b: ax = c − b

Step 2 (Linear Equations)

Divide by a: x = (c−b)/a

Tactic (Linear Equations)

Always isolate x. Check your answer by plugging it back in.

Slope (Linear Functions)

m = (y2 - y1) / (x2 - x1)

Slope-intercept Equation

y = mx + b

Point-slope Equation

y - y1 = m(x - x1)

Find y-intercept (Graphing)

Plot y-intercept = b

Use slope to find next point

e.g., m = 2 → up 2, right 1

Standard Form (Linear Equations in Two Variables)

Ax + By = C

Find x-intercept

Set y = 0, solve for x.

Find y-intercept

Set x = 0, solve for y.

Slope from Standard Form

m = -B/A

Methods to Solve Systems of Linear Equations

1. Substitution, 2. Elimination, 3. Graphing (use Desmos for accuracy)

One Solution (Systems of Linear Equations)

Lines intersect.

No Solution (Systems of Linear Equations)

Parallel lines (same slope, different intercepts).

Infinite Solutions (Systems of Linear Equations)

Same line (identical equations).

Solving Linear Inequalities

Same as equations, but flip the sign if multiplying/dividing by a negative.

Graphing Linear Inequalities

Solid line for ≤ or ≥; Dashed line for < or >.

Shifting a Function Horizontally

f(x) → f(x-h): horizontal shift to the right [(-)→right, (+)→left]

Shifting a Function Vertically

f(x) → f(x) + k: vertical shift up [(-)→down,(+)→up]

Quadratic Functions

y = ax^2 + bx + c

Sum of the Roots (Quadratic Functions)

-b/a

Product of the Roots (Quadratic Functions)

c/a

Vertex of a Parabola

x = -b/(2a)

Systems with nonlinear equations

Substitute or graph to find intersection points.

Equivalent Expressions

Simplify: Combine like terms, factor, or expand.

Rational expressions

Factor numerator and denominator, then cancel common terms.

Ratio

a to b or b to a.

Proportion

Cross-multiply: ad = bc.

Unit conversion

Use conversion factors (e.g., 1 km = 0.62 mi).

Part Formula

Part = Whole × Percent / 100.

Whole Formula

Whole = part × 100 / percentage.

Percentage Formula

Percentage = Part / Whole × 100.

Percent change

Percent change = (New - Old) / Old × 100%.

Mean

Mean = Σx / n.

Median

Middle value when ordered.

Mode

Most frequent value.

Range

Max - min.

Standard deviation

Measure of spread (larger = more spread out).

Line of best fit

Approximates the trend (use linear regression on Desmos).

Correlation (r)

r = 1: Perfect positive linear relationship; r = -1: Perfect negative linear relationship; r = 0: No linear relationship.

Basic probability

P(A) = Favorable outcomes / Total outcomes.

Conditional probability

P(A|B) = P(A and B) / P(B).

Margin of error

MOE ≈ 1 / √n (for 95% confidence).

Observational studies

No intervention (correlation ≠ causation).

Experimental studies

Treatment applied (can infer causation).

Rectangle Area and Perimeter

A = lw, P = 2l + 2w.

Triangle Area

A = 1/2 bh.

Circle Area and Circumference

A = πr², C = 2πr.

Rectangular prism Volume

V = lwh.

Pyramid Volume

V = 1/3 b h.

Cylinder Volume

V = πr²h.

Cone Volume

V = 1/3 πr²h.

Sphere Volume

V = 4/3 πr³.

Pythagorean theorem

a² + b² = c² (right triangles).

Inscribed Angles

Angles formed by two chords in a circle.

Arc length

L = (θ / 360) × 2πr.

Sector area

A = (θ / 360) × πr².