39.7 Elementary Linear System Applications

Elementary Linear System Applications is the process of translating a real-world situation involving two unknown quantities into a system of two linear equations, solving that system using substitution or elimination, and interpreting the resulting ordered pair back within the original context.

Two Unknown Context Quantities

Description

Every application begins by identifying exactly two quantities within a real-world situation whose values are not yet known, but which are related to each other through the conditions described in the situation.

Example Situation

A situation describing the combined cost of two different items purchased in different amounts, where the price of each individual item is unknown, provides exactly two unknown quantities suitable for this method.

System Variable Declaration

Procedure

Each unknown quantity identified in the situation is assigned a distinct variable, along with a clear statement of what that variable represents, including any relevant units.

Example

Letting $x$ represent the price of one item in dollars, and $y$ represent the price of a second item in dollars, clearly declares the two variables needed for the system.

First Context Condition Equation

Procedure

The first relationship described in the situation is translated directly into a linear equation involving both declared variables.

Example

If three units of the first item and two units of the second item together cost sixteen dollars, this condition becomes:

Second Context Condition Equation

Procedure

A second, independent relationship described in the same situation is translated into a second linear equation involving the same two variables.

Example

If one unit of the first item and one unit of the second item together cost seven dollars, this condition becomes:

completing the system alongside the first condition equation.

Application Method Selection

Procedure

Either the substitution method or the elimination method is chosen to solve the resulting system, based on which equation's structure makes solving more direct.

Example

Since the second equation isolates neither variable but has a coefficient of one on both, substitution is convenient here: solving $x+y=7$ for $y$ gives $y=7-x$, ready for substitution into the first equation.

Contextual Ordered Pair Interpretation

Procedure

The algebraic ordered pair produced by solving the system is translated back into the original context, restoring the meaning and units originally assigned to each variable.

Example

Solving the system above produces $(2,5)$, which is interpreted as the first item costing two dollars and the second item costing five dollars.

Application Condition Verification

Procedure

The interpreted values are checked against both original conditions stated in the situation, in their original wording, to confirm the solution makes sense in context.

Example

Three units at two dollars plus two units at five dollars gives sixteen dollars, matching the first condition, and one unit of each gives seven dollars, matching the second condition, confirming the solution is correct.

Contextually Invalid System Result

Description

An algebraically correct solution can still be invalid within the context of the situation, such as when a computed price, length, or count of items is negative or otherwise impossible for the quantity it represents.

Handling This Case

When a solution is contextually invalid, the original equations are reviewed for translation errors, since a properly formed real-world system describing physically meaningful quantities should not produce an impossible result.