BackCollege Algebra: Word Problems Involving Linear Equations and Integer Applications
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Word Problems Involving Linear Equations
Solving for Unknowns Using Systems of Equations
Many real-world problems can be modeled and solved using linear equations. These problems often involve relationships between quantities, such as earnings, lengths, or sales, and require translating words into algebraic expressions.
Key Point 1: Assign variables to unknown quantities and express all relationships in terms of these variables.
Key Point 2: Set up equations based on the information given, then solve for the variables using algebraic methods.
Key Point 3: Check your solutions by substituting them back into the original context to ensure they make sense.
Example 1: Earnings Problem
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 x = amount Jim earned.
Bill earned 3 times as much: 3x.
Tom earned $5 more than Jim: x + 5.
Total earnings: $90.
Equation:
Simplify and solve:
Jim: $17
Bill: $51
Tom: $22
Application: This type of problem is common in budgeting, payroll, and resource allocation scenarios.
Integer Applications in Geometry
Consecutive Integers and Perimeter Problems
Problems involving consecutive integers often appear in geometry, especially when dealing with perimeters or other measurements. Consecutive integers are numbers that follow each other in order, differing by 1.
Key Point 1: Represent three consecutive integers as x, x+1, and x+2.
Key Point 2: The perimeter of a triangle is the sum of its three sides.
Example 2: Triangle Side Lengths
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 the sides be x, x+1, and x+2.
Perimeter:
Simplify and solve:
Sides: 5 ft, 6 ft, 7 ft
Application: This approach is useful for problems involving measurements, such as fencing, construction, or design.
Word Problems Involving Price and Quantity
Setting Up and Solving Equations from Sales Scenarios
Sales and revenue problems often require setting up equations based on price changes and total sales. These problems test the ability to translate a scenario into algebraic expressions and solve for unknowns.
Key Point 1: Assign variables to unknown quantities, such as the number of items sold at different prices.
Key Point 2: Write an equation for the total number of items and another for the total revenue.
Example 3: Lemonade Stand Problem
Problem: A child is selling lemonade. She started by charging $2 per cup, but later 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? Let x be the number of cups sold at $2 per cup.
Let x = number of cups sold at $2.
Then 50 - x = number of cups sold at $3.
Total revenue:
Simplify and solve:
Cups sold at $2: 26
Cups sold at $3: 24
Expression for cups sold at $3:
Application: This type of problem is common in business, economics, and inventory management.
