Using Linear Equations to Solve Problems

Vocabulary and core concepts for setting up and solving linear equations from worded problems, covering key steps, algebraic modeling, financial relationships, and consecutive integers.

Pronumeral

A letter used in algebra to stand for an unknown number in a worded problem.

Linear equation

An algebraic statement containing an equals sign and unknown variables to the first power, written and solved to find unknown values in worded situations.

Four steps for solving a word problem

The standard procedure: 1) Define a pronumeral for the unknown numbers; 2) Write an equation to describe the situation; 3) Solve the equation by inspection or systematically; 4) Answer the original question, including the correct units.

Solving by inspection

Determining the solution to an equation mentally by observing which value makes the equation true.

Solving systematically

Solving an equation step-by-step by performing inverse operations on both sides to isolate the pronumeral.

Initial fee

A fixed upfront charge that does not depend on the number of hours or items, added to the variable rate when setting up a linear cost equation (such as the $\$10$ fee for hiring a bike).

Hourly rate

A variable charge applied per hour of hire or service that is multiplied by the number of hours ($h$) in a linear equation (such as $\$8$ per hour).

Expressing multiple unknowns in terms of one pronumeral

A method where one unknown is defined as a variable (such as $x$) and other related unknowns are expressed algebraically relative to it (such as $x + 68$).

Consecutive integers

Integers that follow each other in order without gaps, represented algebraically as $x$, $x + 1$, $x + 2$, etc.

Revenue

The total amount of money collected from selling goods or services, calculated as the unit selling price multiplied by the number of items sold ($R = 24n$).

Start-up cost

An initial fixed production expenditure incurred prior to manufacturing (such as $\$7200$) that is added to per-unit costs to determine total cost.

Profit

The net financial gain remaining after expenses are deducted, calculated by the formula $\text{Profit} = \text{revenue} - \text{costs}$.