Math Problem Statement
6 pencils and 5 crayons are 32.0 inches. 7 pencils and 6 crayons are 37.7 inches. Each pencil is the same length as the other pencils, and each crayon is the same length as the other crayons. What is the length of 1 pencil? What is the length of 1 crayon?
Solution
We can solve this problem using a system of equations. Let the length of one pencil be inches and the length of one crayon be inches.
From the problem, we have the following two equations:
- (Equation 1)
- (Equation 2)
Step 1: Multiply both equations to eliminate one variable
To eliminate one variable, we'll multiply both equations so the coefficients of (the number of crayons) are the same. We'll multiply Equation 1 by 6 and Equation 2 by 5:
Step 2: Subtract the two equations
Now subtract the second equation from the first:
So, the length of one pencil is 3.5 inches.
Step 3: Substitute the value of into one of the original equations
Substitute into Equation 1:
So, the length of one crayon is 2.2 inches.
Final Answer:
- The length of one pencil is 3.5 inches.
- The length of one crayon is 2.2 inches.
Would you like more details or have any questions?
Here are 5 related questions:
- What method would you use to solve a similar problem with different coefficients?
- Can you verify the solution by substituting both values into the second equation?
- How would the solution change if the sum of the lengths in both equations was larger?
- What if the system of equations had decimals in the coefficients?
- How could graphing help in visualizing the solution?
Tip: Always check your solution by substituting the values back into both original equations to ensure they satisfy both conditions.
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Math Problem Analysis
Mathematical Concepts
Algebra
System of Linear Equations
Formulas
Linear equation formula: ax + by = c
Theorems
Substitution Method
Elimination Method
Suitable Grade Level
Grades 8-10
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