SAT Math 1: Linear Equations and InequalitiesSAT Math 1: Linear Equations and Inequalities

Linear Equations and Inequalities

Goal: Solve linear equations and inequalities and interpret slope and intercept in context.

THE SHORT VERSION

Linear relationships appear everywhere on the SAT Math section: solving for a variable, reading a line's slope and intercept, and translating word problems into equations.

1 Solving Linear Equations

Do the same operation to both sides. Distribute first, collect like terms, then isolate the variable. Fractions: multiply both sides by the least common denominator.

One solution (x = 5), no solution (a false statement such as 3x + 2 = 3x + 5), or infinitely many solutions (a true statement such as 4(x + 3) = 4x + 12).

Example: 3(x − 2) = 2x + 5 → 3x − 6 = 2x + 5 → x = 11.

2 Lines: Forms, Slope, and Meaning

Form Equation What you read off

Slope-intercept y = mx + b Slope m, y-intercept b

Point-slope y − y₁ = m(x − x₁) Slope m, point (x₁, y₁)

Standard Ax + By = C Slope = −A/B; x- and y-intercepts via setting one variable to 0

Slope = (y₂ − y₁)/(x₂ − x₁) = rise/run.

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals (2 and −1/2). In context: slope is the rate of change per unit of x; the y-intercept is the starting value (when x = 0). Example: a plumber charges a $45 call fee plus $30 per hour: C = 30h + 45. The 30 is the hourly rate; the 45 is the flat fee. Number line for x at least 3, and the line y = 2x - 3.

3 Linear Inequalities

Solve like an equation, but flip the inequality sign when multiplying or dividing by a negative number. Example: −2x + 7 > 15 → −2x > 8 → x < −4.

Wording: "at least" means ≥; "at most" and "no more than" mean ≤; "fewer than" means <; "more than" means >. Absolute value: |x − a| < b means a − b < x < a + b. |x − a| > b means x < a − b or x > a + b.

A solution set on a number line uses an open circle for < and >, a closed circle for ≤ and ≥.

4 Translating Words into Equations

Phrase Math

is, was, equals =

more than / increased by +

less than ("5 less than x") x − 5

of (with a fraction or percent) multiply

per divide (rate)

twice a number, then 3 more 2x + 3

5 Check Yourself

Q: How many solutions does 4(x + 3) = 4x + 12 have? A: Infinitely many.

Q: A line has slope 3 and passes through (2, 7). What is its y-intercept? A: y = 3x + b, so 7 = 6 + b and b = 1. Q: If 2x + 5 ≤ 17, what is the greatest value of x? A: x ≤ 6, so 6.

6 Words to know

Slope

Rate of change of a line.

y-intercept

Value of y when x = 0.

Linear inequality

Relationship using <, >, ≤, or ≥.

Absolute value

Distance from zero (or from a number).

Negative reciprocal

Flip the fraction and change the sign; gives the perpendicular slope.

STRATEGY Plug answer choices back into the equation if solving algebraically is slow. TRAP Reading the question's last sentence: it may ask for 2x or x + y, not x.

SAT Math Guide 1 · original study notes