To find the length of one pencil and one crayon, let's set up a system of equations based on the information given.

Let:

• $p$ represent the length of one pencil (in inches)
• $c$ represent the length of one crayon (in inches)

The problem gives us two equations:

1. $6p + 5c = 32.0$
2. $7p + 6c = 37.7$

We can solve this system using either substitution or elimination. I'll use elimination here:

Step 1: Eliminate one variable

To eliminate $c$, we can multiply the first equation by 6 and the second equation by 5, so the coefficients of $c$ will be the same in both equations.

1. $6(6p + 5c) = 6 \cdot 32.0$
   $36p + 30c = 192.0$

2. $5(7p + 6c) = 5 \cdot 37.7$
   $35p + 30c = 188.5$

Step 2: Subtract the equations

Now, subtract the second equation from the first to eliminate $c$:

$(36p + 30c) - (35p + 30c) = 192.0 - 188.5$

This simplifies to:

$p = 3.5$

So, the length of one pencil, $p$, is 3.5 inches.

Step 3: Substitute to find $c$

Now that we know $p = 3.5$, substitute it back into one of the original equations to solve for $c$. Using the first equation:

$6(3.5) + 5c = 32.0$

$21.0 + 5c = 32.0$

Subtract 21.0 from both sides:

$5c = 11.0$

Divide by 5:

$c = 2.2$

So, the length of one crayon, $c$, is 2.2 inches.

Final Answer

• Length of one pencil: 3.5 inches
• Length of one crayon: 2.2 inches

Would you like further details on this solution or have any questions?

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Related Questions for Practice:

1. If 10 pencils and 8 crayons were used, what would be the total length?
2. What if the lengths were in centimeters instead? How would you convert?
3. How would the solution change if the total lengths were given in different units?
4. Can you set up a similar problem with two unknowns but using weights instead of lengths?
5. How would you solve this problem using substitution instead of elimination?

Tip: When solving systems of equations, always double-check by substituting your values back into the original equations to ensure accuracy.