Mastering the Heart of Algebra: SAT Math Structure and Strategy

Linear Equation

An equation that expresses a relationship where a single variable moves at a constant rate, typically in the form of ax + b = c.

One Solution

Occurs when the variable terms on both sides of an equation have different coefficients.

No Solution

Occurs when the variable terms are identical but the constant terms are different, indicating parallel lines that do not intersect.

Infinite Solutions

Occurs when both the variable terms and the constants are identical, meaning every real number is a solution.

Slope-Intercept Form

A form of a linear equation, written as y = mx + b, where m is the slope and b is the y-intercept.

Standard Form

A linear equation format written as Ax + By = C, commonly used for problems involving multiple item types.

Point-Slope Form

A form written as y - y₁ = m(x - x₁), useful for equations with a slope and a specific point.

Rate of Change

The slope (m) of a linear function, indicating how y changes in relation to x.

Y-Intercept

The value of y when x = 0, also referred to as the initial value. Identified by the term b in the slope-intercept form.

X-Intercept

The value of x when y = 0, often used to determine when a total amount reaches zero.

HOY VUX

A mnemonic for remembering the characteristics of horizontal (HOY) and vertical (VUX) lines regarding their slopes.

Substitution Method

A method for solving systems of equations by isolating one variable and substituting it into another equation.

Elimination Method

A technique for solving systems of equations that involves adding or subtracting equations to eliminate a variable.

Solution to a System

The ordered pair (x, y) that satisfies both equations in a system of linear equations.

Number of Solutions

Determined by comparing the slopes and intercepts of two linear equations.

Strict Inequality

An inequality that uses < or > signs, indicating that the boundary line should be dashed when graphed.

Inclusive Inequality

An inequality using ≤ or ≥ signs, indicating that the boundary line should be solid when graphed.

Shaded Region

The area on a graph representing the solution set for a linear inequality.

Common Mistake: Negative Flip

A frequent error where students neglect to flip the inequality sign when dividing by a negative number.

Perpendicular Slopes

Slopes that are negative reciprocals of one another; if m = 2, the perpendicular slope is -1/2.

Distribution Error

Mistakes made when distributing, such as failing to distribute the negative sign in an equation.

Identical Lines

Occurs in a system of equations when both slopes and intercepts are equal, indicating infinite solutions.

Parallel Lines

Lines that have equal slopes but different y-intercepts, resulting in no solutions.

Linear Inequality in One Variable

An inequality that represents a range of values rather than a specific solution.

Graphing Linear Inequalities

Involves using dashed or solid lines and shading to indicate the solution region on a coordinate plane.

System of Linear Equations

A set of two or more linear equations with the same variables.

Identifying Solutions Graphically

Finding the point of intersection of two lines to determine the solution to a system of equations.