Linear Systems and Relations Vocabulary

Vocabulary and definition flashcards covering linear relations, linear systems, graphical solutions, and algebraic modeling based on lecture notes.

Relation

A statement that describes how two variables are related to each other.

Linear Relation

A relation in which two variables relate such that their graph forms a straight line.

Representations of a Linear Relation

The three methods used to represent a linear relation: an equation, a table of values, and a graph.

System of Equations

A set of two or more equations used together to model a problem.

Linear System

A system of equations in which all of the equations are linear.

Linear System in Two Unknowns

A linear system consisting of two or more linear equations that involve two variables, such as $2x + 3y = 8$ and $5x - 2y = 1$.

Solution to a Linear System

An ordered pair $(x, y)$ that satisfies all equations in the system, corresponding graphically to the point of intersection.

Left Side Equals Right Side ($LS = RS$)

The mathematical verification condition achieved when substituting a candidate solution point into an equation yields an equivalent result on both sides.

CanFit Fitness Club Charge Model

The linear relation $y = 20x + 150$, where $x$ represents the number of months and $y$ represents total charges incorporating a \text{\\$150} initial fee and a \text{\\$20} monthly fee.

Mario's Daily Earnings Model

The linear relation $y = 0.12x + 80$, where $x$ is sales and $y$ is daily earnings based on an \text{\\$80} base pay plus $12\%$ commission.

IT Plus vs. Techies Inc. Internet Models

The equations $y = 25$ (IT Plus flat rate) and $y = x + 10$ (Techies Inc. rate), where IT Plus is the more cost-effective option for $18\,\text{h}$ of monthly usage (\text{\\$25} compared to \text{\\$28}).

Airplane Rental Cost Comparison Model

The system of equations $y = 50x + 300$ (flying own plane) and $y = 100x$ (renting from flying club), which yields an equal monthly cost at $x = 6\,\text{hours}$.

Slope-Intercept Graphical Solution Method

A method for solving a linear system by converting equations to slope-intercept form, such as $3x + y = 5 \rightarrow y = -3x + 5$ and $-x + 3y = -15 \rightarrow y = \frac{1}{3}x - 5$, to find their point of intersection at $(3, -4)$.

Table of Values Solution Method

A technique to find the point of intersection $(3, -1)$ by evaluating tables of values for the system $2x + y = 5$ and $y = \frac{2}{3}x - 3$.