BackCollege Algebra Study Notes: Word Problems and Linear Equations
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Word Problems Involving Linear Equations
Solving Word Problems with Systems of Equations
Word problems in College Algebra often require translating real-life scenarios into algebraic equations. These equations can then be solved to find unknown quantities. The following examples illustrate common types of word problems involving linear equations and systems.
Key Point 1: Identify unknowns and assign variables to represent them.
Key Point 2: Translate the relationships described in the problem into algebraic equations.
Key Point 3: Use substitution or elimination methods to solve the equations.
Example 1: Earnings Comparison
Problem: One weekend Bill earned 3 times as much as Jim. Tom earned $5 more than Jim. In all, they earned $90. How much did each earn?
Let be the amount Jim earned.
Bill earned:
Tom earned:
Total earned:
Equation:
Jim:
Bill:
Tom:
Application: This type of problem is common in budgeting, payroll, and financial planning.
Example 2: Consecutive Integer Sides of a Triangle
Problem: The lengths of the sides of a triangle are represented by three consecutive integers. If the perimeter of the triangle is 18 feet, find the lengths of its sides.
Let be the smallest integer.
Sides: , ,
Perimeter:
Equation:
Sides: , ,
Application: Problems involving consecutive integers are useful for understanding sequences and series.
Example 3: Lemonade Stand Revenue Problem
Problem: A child is selling lemonade. She started by charging $2 per cup, then increased the price to $3 per cup. After selling all 50 cups, she had earned $124. How many cups did she sell at $2, and how many at $3? Can you come up with an expression for the number of cups sold at $3?
Let be the number of cups sold at .
Let be the number of cups sold at .
Total cups:
Total revenue:
System of Equations:
Solving:
From the first equation:
Substitute into the second equation:
Cups sold at $2: 26
Cups sold at $3: 24
Expression for cups sold at $3: (from )
Application: This type of problem is useful for understanding systems of equations and their applications in business and sales.
Summary Table: Word Problem Types and Solution Methods
Problem Type | Variables Used | Equation(s) | Solution Method |
|---|---|---|---|
Earnings Comparison | Jim's earnings () | Combine like terms, solve for | |
Consecutive Integer Sides | Smallest side () | Combine like terms, solve for | |
Lemonade Stand Revenue | Cups at x$), cups at y$) |
| Substitution or elimination |
Additional info: These problems are foundational for understanding algebraic modeling, systems of equations, and problem-solving strategies in College Algebra.
