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?

We can solve this problem using a system of equations. Let the length of one pencil be \( p \) inches and the length of one crayon be \( c \) inches.

From the problem, we have the following two equations:

1. \( 6p + 5c = 32.0 \)  (Equation 1)
2. \( 7p + 6c = 37.7 \)  (Equation 2)

### Step 1: Multiply both equations to eliminate one variable

To eliminate one variable, we'll multiply both equations so the coefficients of \( c \) (the number of crayons) are the same. We'll multiply Equation 1 by 6 and Equation 2 by 5:

\[
6(6p + 5c) = 6(32.0) \quad \Rightarrow \quad 36p + 30c = 192.0
\]
\[
5(7p + 6c) = 5(37.7) \quad \Rightarrow \quad 35p + 30c = 188.5
\]

### Step 2: Subtract the two equations

Now subtract the second equation from the first:

\[
(36p + 30c) - (35p + 30c) = 192.0 - 188.5
\]
\[
p = 3.5
\]

So, the length of one pencil is 3.5 inches.

### Step 3: Substitute the value of \( p \) into one of the original equations

Substitute \( p = 3.5 \) into Equation 1:

\[
6(3.5) + 5c = 32.0
\]
\[
21.0 + 5c = 32.0
\]
\[
5c = 11.0
\]
\[
c = 2.2
\]

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:
1. What method would you use to solve a similar problem with different coefficients?
2. Can you verify the solution by substituting both values into the second equation?
3. How would the solution change if the sum of the lengths in both equations was larger?
4. What if the system of equations had decimals in the coefficients?
5. 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.

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?

Learn how to solve a system of linear equations to determine the length of pencils and crayons. This problem involves algebraic concepts such as substitution and elimination methods, making it suitable for students in Grades 8-10.

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