SAT Math Algebra: A Conceptual Guide to Lines, Systems, and Inequalities

Linear equation in 1 variable

An equation where the variable appears only to the first power; a typical form is ax + b = c.

Balance idea

The principle that an equation is like a balanced scale, so whatever you do to one side, you must do to the other.

Distribute

Apply multiplication to every term inside parentheses, such as turning 3(x - 2) into 3x - 6.

Clear denominators

Remove fractions by multiplying every term in an equation by a common denominator.

No solution

A case where simplifying leads to a false statement, such as 0 = 5, so no value makes the equation true.

Infinitely many solutions

A case where simplifying leads to a true statement, such as 0 = 0, so every value satisfies the equation.

Linear equation in 2 variables

An equation of the form ax + by = c whose solutions are ordered pairs (x, y).

Standard form

A way to write a line as ax + by = c; it is useful for elimination and showing integer relationships.

Slope-intercept form

A way to write a line as y = mx + b, where m is the slope and b is the y-intercept.

Point-slope form

A way to write a line as y - y1 = m(x - x1); it is useful when you know one point and the slope.

x-intercept

The point where a line crosses the x-axis; find it by setting y = 0.

y-intercept

The point where a line crosses the y-axis; find it by setting x = 0; in context, it often represents an initial value.

Linear function

A function with a constant rate of change whose graph is a straight line, often written as f(x) = mx + b.

Function

A relationship in which each input x produces exactly one output y.

Slope

The rate of change of a line, computed as (y2 - y1)/(x2 - x1), also called rise over run.

Vertical line

A line with equation x = a that has undefined slope and is not a function of x.

System of linear equations

Two linear equations considered together; the solution is the ordered pair that satisfies both equations.

Substitution

A method for solving a system by solving one equation for a variable and plugging that expression into the other equation.

Elimination

A method for solving a system by adding or subtracting equations to cancel one variable.

Parallel lines

Lines with the same slope but different intercepts; in a system, they never intersect, so there is no solution.

Linear inequality

A comparison using

Sign-flip rule

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Compound inequality

A statement with two inequalities joined together, such as -1 < 2x + 3 <= 7, solved by applying operations to all three parts.

Boundary line

In a two-variable inequality, the line formed by replacing the inequality sign with =; it separates the plane into regions.

Test point

A point, often (0, 0), substituted into an inequality to determine which side of the boundary line is the solution region.