Math Problem Statement
Determine the prices of a winter hat and a cap based on the given equations involving their costs.
Solution
To solve the problem:
-
Define the variables:
- Let the price of the winter hat =
- Let the price of the cap =
-
From the first equation:
Simplify to:
2H + C = 18 \tag{1} -
From the second equation:
C = H + H \tag{2}
Simplify to:
-
Substitute from equation (2) into equation (1):
Combine terms:
Solve for :
-
Use in equation (2) to find :
Final Solution:
- The winter hat costs $4.50.
- The cap costs $9.00.
Would you like a breakdown of the steps or further explanation?
Here are 5 related questions to practice:
- If the price of the hat were $5, what would the cap's price be?
- How would the solution change if the total was $20 instead of $18?
- What happens if the cap costs $10? Can you solve for the hat's price?
- Could you rewrite the equations for three hats and one cap summing to $27?
- Can you use a graphical method to solve the problem instead?
Tip: Always double-check your substitution in equations to ensure accuracy!
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Formulas
Substitution method in solving systems of linear equations
Theorems
Properties of equality
Suitable Grade Level
Grades 6-8