Algebra y Trigonometría CECC: Applications of Linear Equations

Learning Objectives

Identify key words and phrases, translate sentences to mathematical equations, and develop strategies to solve problems.
Solve word problems involving relationships between numbers.
Solve geometry problems involving perimeter.
Solve percent and money problems including simple interest.
Set up and solve uniform motion problems.

Key Words, Translation, and Strategy

Algebra simplifies the process of solving real-world problems. This is done by using letters to represent unknowns, restating problems in the form of equations, and offering systematic techniques for solving those equations. To solve problems using algebra, first translate the wording of the problem into mathematical statements that describe the relationships between the given information and the unknowns. Usually, this translation to mathematical statements is the difficult step in the process. The key to the translation is to carefully read the problem and identify certain key words and phrases.

Key Words	Translation
Sum, increased by, more than, plus, added to, total	+
Difference, decreased by, subtracted from, less, minus	−
Product, multiplied by, of, times, twice	*
Quotient, divided by, ratio, per	/
Is, total, result	=

Here are some examples of translated key phrases.

Key Phrases	Translation
The sum of a number and 7.	$x + 7$
Seven more than a number.	$x + 7$
The difference of a number and 7.	$x - 7$
Seven less than a number.
Seven subtracted from a number.
The product of 2 and a number.	$2 x$
Twice a number.	$2 x$
One-half of a number.	$1 2 x$
The quotient of a number and 7.	$x / 7$

When translating sentences into mathematical statements, be sure to read the sentence several times and identify the key words and phrases.

Example 1: Translate: Four less than twice some number is 16.

Solution: First, choose a variable for the unknown number and identify the key words and phrases.

Let x represent the unknown indicated by “some number.”

Remember that subtraction is not commutative. For this reason, take care when setting up differences. In this example, 4−2x=16 is an incorrect translation.

Answer: 2x−4=16

It is important to first identify the variable—let x represent…—and state in words what the unknown quantity is. This step not only makes your work more readable but also forces you to think about what you are looking for. Usually, if you know what you are asked to find, then the task of finding it is achievable.

Example 2: Translate: When 7 is subtracted from 3 times the sum of a number and 12, the result is 20.

Solution: Let n represent the unknown number.

Answer: 3(n+12)−7=20

To understand why parentheses are needed, study the structures of the following two sentences and their translations:

“3 times the sum of a number and 12”	$3 (n + 12)$
“the sum of 3 times a number and 12”	$3 n + 12$

The key is to focus on the phrase “3 times the sum.” This prompts us to group the sum within parentheses and then multiply by 3. Once an application is translated into an algebraic equation, solve it using the techniques you have learned.

Guidelines for Setting Up and Solving Word Problems

Step 1: Read the problem several times, identify the key words and phrases, and organize the given information.

Step 2: Identify the variables by assigning a letter or expression to the unknown quantities.

Step 3: Translate and set up an algebraic equation that models the problem.

Step 4: Solve the resulting algebraic equation.

Step 5: Finally, answer the question in sentence form and make sure it makes sense (check it).

For now, set up all of your equations using only one variable. Avoid two variables by looking for a relationship between the unknowns.

Problems Involving Relationships between Real Numbers

We classify applications involving relationships between real numbers broadly as number problems. These problems can sometimes be solved using some creative arithmetic, guessing, and checking. Solving in this manner is not a good practice and should be avoided. Begin by working through the basic steps outlined in the general guidelines for solving word problems.

Example 3: A larger integer is 2 less than 3 times a smaller integer. The sum of the two integers is 18. Find the integers.

Solution:

Identify variables: Begin by assigning a variable to the smaller integer.

Use the first sentence to identify the larger integer in terms of the variable x: “A larger integer is 2 less than 3 times a smaller.”

Set up an equation: Add the expressions that represent the two integers, and set the resulting expression equal to 18 as indicated in the second sentence: “The sum of the two integers is 18.”

Solve: Solve the equation to obtain the smaller integer x.

Back substitute: Use the expression 3x−2 to find the larger integer—this is called back substituting.

Answer the question: The two integers are 5 and 13.

Check: 5 + 13 = 18. The answer makes sense.

Example 4: The difference between two integers is 2. The larger integer is 6 less than twice the smaller. Find the integers.

Solution: Use the relationship between the two integers in the second sentence, “The larger integer is 6 less than twice the smaller,” to identify the unknowns in terms of one variable.

Since the difference is positive, subtract the smaller integer from the larger.

Solve.

Use 2x − 6 to find the larger integer.

Answer: The two integers are 8 and 10. These integers clearly solve the problem.

It is worth mentioning again that you can often find solutions to simple problems by guessing and checking. This is so because the numbers are chosen to simplify the process of solving, so that the algebraic steps are not too tedious. You learn how to set up algebraic equations with easier problems, so that you can use these ideas to solve more difficult problems later.

Example 5: The sum of two consecutive even integers is 46. Find the integers.

Solution: The key phrase to focus on is “consecutive even integers.”

Add the even integers and set them equal to 46.

Solve.

Use x + 2 to find the next even integer.

Answer: The consecutive even integers are 22 and 24.

It should be clear that consecutive even integers are separated by two units. However, it may not be so clear that odd integers are as well.

Example 6: The sum of two consecutive odd integers is 36. Find the integers.

Solution: The key phrase to focus on is “consecutive odd integers.”

Add the two odd integers and set the expression equal to 36.

Solve.

Use x + 2 to find the next odd integer.

Answer: The consecutive odd integers are 17 and 19.

The algebraic setup for even and odd integer problems is the same. A common mistake is to use x and x + 3 when identifying the variables for consecutive odd integers. This is incorrect because adding 3 to an odd number yields an even number: for example, 5 + 3 = 8. An incorrect setup is very likely to lead to a decimal answer, which may be an indication that the problem was set up incorrectly.

Example 7: The sum of three consecutive integers is 24. Find the integers.

Solution: Consecutive integers are separated by one unit.