PSAT 10 Math Algebra: Linear Relationships, Systems, and Inequalities

Linear equation in one variable

An equation where the variable appears only to the first power and is not multiplied by itself.

Standard form of linear equation

The typical form is $ax + b = c$ where $x$ is the variable, $a$ is the coefficient, and $b$ and $c$ are constants.

Inverse operations

Operations that undo each other; used to isolate the variable in an equation.

No solution

Occurs when a simplified equation results in a false statement (e.g., $0 = 5$).

Infinitely many solutions

Occurs when a simplified equation results in a true statement (e.g., $0 = 0$).

Slope-intercept form

Form of a linear equation written as $y = mx + b$, emphasizing the slope $m$ and the $y$-intercept $b$.

Point-slope form

Form of a linear equation written as $y - y_1 = m(x - x_1)$, used when a point and slope are known.

Rate of change

The slope $m$ of a linear function, representing the change in output for a change in input.

Graph of a linear equation in two variables

A straight line in the coordinate plane, representing all solutions of the equation.

System of linear equations

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

Substitution method

A technique for solving systems of equations by solving one equation for one variable and substituting into the other.

Elimination method

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

Linear inequality

An inequality that resembles a linear equation but uses $, \, \neq, \, \le, \ge$ signs.

Shading in graphing inequalities

The part of the graph that includes all solutions to an inequality, determined by the inequality sign.

Test point in inequalities

A point used to determine which side of a boundary line is part of the solution set for an inequality.

Compound inequality

An inequality combining two inequalities joined by 'and' or 'or', solved together.

Solid boundary line

Used when graphing inequalities that include the boundary points, represented by $\le$ or $\ge$.

Dashed boundary line

Used when graphing inequalities that do not include the boundary points, represented by $<$ or $>$.

Intercepts

Points where a graph intersects the axes, specifically $x$-intercept (when $y=0$) and $y$-intercept (when $x=0$).

Coefficient

The numerical factor multiplying the variable in an equation.

Simple linear equation

An uncomplicated linear equation, often of the form $ax + b = c$ that may require minimal manipulation.

Modeling with linear functions

Using equations to represent real-world situations by establishing relationships between variables.

Parallel lines in a system

Lines that never intersect, indicating the system has no solution.

Same line in a system

Lines that coincide, indicating the system has infinitely many solutions.

Error in solving equations

Common mistakes made during calculations or simplifications when solving mathematical equations.

Functional notation

A way of expressing functions as $f(x)$ instead of using $y$, emphasizing function output.

Interval notation

A mathematical notation for representing subsets of real numbers, often used in solutions to inequalities.