- Create and use linear equations and formulas to solve practical problems involving unknown quantities, dimensions, and distances.
Setting up a Linear Equation to Solve a Real-World Application
Many real-world applications can be modeled by linear equations. For example: Josh is hoping to get an A in his college algebra class. He has scores of [latex]75, 82, 95, 91,[/latex] and [latex]94[/latex] on his first five tests. Only the final exam remains, and the maximum of points that can be earned is [latex]100[/latex]. Is it possible for Josh to end the course with an A? A linear equation will give Josh his answer.
To set up a linear equation that models a real-world situation, we must first determine the known quantities and define the unknown quantity as a variable. Then, we interpret the words as mathematical expressions using mathematical symbols. If a quantity is independent of a variable, we usually just add or subtract it according to the problem.
When dealing with real-world applications, there are certain expressions that we can translate directly into math. The table below lists some common verbal expressions and their equivalent mathematical expressions.
| Verbal | Translation to Math Operations |
|---|---|
| One number exceeds another by a | [latex]x,\text{ }x+a[/latex] |
| Twice a number | [latex]2x[/latex] |
| One number is a more than another number | [latex]x,\text{ }x+a[/latex] |
| One number is a less than twice another number | [latex]x,2x-a[/latex] |
| The product of a number and a, decreased by b | [latex]ax-b[/latex] |
| The quotient of a number and the number plus a is three times the number | [latex]\Large\frac{x}{x+a}\normalsize =3x[/latex] |
| The product of three times a number and the number decreased by b is c | [latex]3x\left(x-b\right)=c[/latex] |
In this case, a known cost, such as [latex]$0.10[/latex] per mile, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write [latex]0.10x[/latex] to model the portion of the weekly cost generated by miles driven. This expression represents a variable cost because it changes according to the number of miles driven. There is a flat fee of [latex]$150[/latex] to rent the car, independent of the number of miles driven. In applications involving costs, amounts such as this flat fee that do not change are often called fixed costs.
In this example, we identified the unknown quantity as the number of miles driven and assigned it the variable [latex]x[/latex]. Next, we identified the known quantities and translated the given information into an equation that models the total weekly cost. The equation [latex]C=150+0.10x[/latex] shows the relationship between the fixed cost of [latex]$150[/latex] to rent the car and the additional cost of [latex]$0.10[/latex] per mile driven. With this model, we can answer questions like how much it would cost to drive [latex]500[/latex] miles in a week or how many miles you could drive on a [latex]$375[/latex] budget.
- Identify known and unknown quantities.
- Assign a variable to represent the unknown quantity.
- If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
- Write an equation interpreting the words in the problem as mathematical operations.
- Solve the equation, check to be sure your answer is reasonable, and give the answer using the language and units of the original situation.
Then, find the two unknown values.
Step 1: Define the variable.
- Let [latex]x[/latex] equal the first number.
- Since the second number exceeds the first by [latex]17[/latex], we write the second number as [latex]x+17[/latex].
Step 2: Set up the equation.
The sum of the two numbers is [latex]31[/latex], which leads us to the equation:
Step 3: Simplify and solve the equation.
- The first number is [latex]x=7[/latex].
- The second number would then be [latex]x+17=7+17=24[/latex].