BackUsing Equations to Solve Word Problems: Translating and Solving in College Algebra
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Section 2.4: Using Equations to Solve Word Problems
Introduction
This section focuses on translating real-world word problems into algebraic equations and solving them. Mastering these skills is essential for success in College Algebra, as it enables students to model and solve practical problems using mathematical reasoning.
Problem Solving Steps
General Approach to Word Problems
Step 1: Read the problem carefully. Identify what is being asked and the information provided.
Step 2: Assign variables. Choose variables to represent unknown quantities.
Step 3: Translate the problem into an equation. Use the relationships described in the problem to write an equation using your variables.
Step 4: Solve the equation. Use algebraic methods to find the value of the unknowns.
Step 5: Check your answer. Substitute your solution back into the original context to verify it makes sense.
Step 6: Write a complete answer. Clearly state your final answer in the context of the problem.
Symbolic Equivalents of English Phrases
Translating English to Mathematical Expressions
Many word problems use specific phrases that correspond to mathematical operations. Recognizing these phrases is key to writing correct equations.
English Phrase | Mathematical Symbol/Expression |
|---|---|
Sum, increased by, more than, added to | + |
Difference, decreased by, less than, subtracted from | - |
Product, multiplied by, of | × or ( ) |
Quotient, divided by, out of | ÷ or fraction bar |
Translating Phrases into Equations
From English Sentences to Algebraic Equations
To translate a sentence into an equation, identify the relationship between quantities and assign variables.
Example: "Amanda is three times older than her brother."
If x is Amanda's age and b is her brother's age, then:
Examples of Translating and Solving Equations
Example 1: Rectangle Length Problem
Problem: The length of the rectangle is 8 feet shorter than double its width.
Let w = width of the rectangle.
Length =
Note: The 8 is subtracted from .
Example 2: Triangle Side Lengths
Problem: A triangle has two sides that are equal in length and a third side that is 3 inches shorter than one and one-half times the length of the equal sides. The perimeter is 28.5 inches.
Let x = length of the equal sides.
Third side =
Perimeter equation:
Simplify:
Solve:
Third side:
Check:
Example 3: Parking Garage Fee Problem
Problem: The Smithfield City Parking Garage charges $6 for the first hour and $3.50 for each additional hour. How much does it cost to park for 19 hours?
Let x = number of additional hours parked (after the first hour).
Equation:
For 19 hours: (since the first hour is $6, the rest are additional)
Calculation:
Answer: $69 for 19 hours of parking.
Summary Table: Translating English to Algebraic Expressions
English Phrase | Algebraic Translation | Example |
|---|---|---|
"a number increased by 5" | If x is the number, then | |
"twice a number" | If x is the number, then | |
"the difference between a number and 7" | If x is the number, then | |
"the quotient of a number and 4" | If x is the number, then |
Key Takeaways
Translating word problems into equations is a foundational skill in algebra.
Recognize common English phrases and their mathematical equivalents.
Follow a systematic approach: read, assign variables, translate, solve, check, and answer.
Practice with real-world examples to build confidence and proficiency.
