https://file.aiquickdraw.com/mathbot/file/1730081832340lehskapj.jpg

To solve this problem, let’s define:

- $ p $ as the cost of one pen.
- $ r $ as the cost of one ruler.

We’re given two equations based on the information:

1. $ 8p + 4r = 116 $
2. $ 2p + 6r = 54 $

Our goal is to find the cost of $ 30 $ pens and $ 30 $ rulers, which would be $ 30p + 30r $.

### Step 1: Simplify the Equations
Let’s simplify the equations to make them easier to solve.

**Equation 1:** $ 8p + 4r = 116 $

Divide by $ 4 $:
\[
2p + r = 29
\]

**Equation 2:** $ 2p + 6r = 54 $

Divide by $ 2 $:
\[
p + 3r = 27
\]

Now we have:
1. $ 2p + r = 29 $
2. $ p + 3r = 27 $

### Step 2: Solve for $ p $ and $ r $
We can use substitution or elimination. Let’s use substitution here.

From the second equation $ p + 3r = 27 $:
\[
p = 27 - 3r
\]

Substitute this expression for $ p $ in the first equation:
\[
2(27 - 3r) + r = 29
\]

Expanding and solving for $ r $:
\[
54 - 6r + r = 29
\]
\[
54 - 5r = 29
\]
\[
5r = 25
\]
\[
r = 5
\]

Now substitute $ r = 5 $ back into the equation $ p = 27 - 3r $:
\[
p = 27 - 3(5)
\]
\[
p = 27 - 15
\]
\[
p = 12
\]

So, the cost of one pen ($ p $) is $ 12 $, and the cost of one ruler ($ r $) is $ 5 $.

### Step 3: Find the Cost of 30 Pens and 30 Rulers
Now, we need to calculate $ 30p + 30r $:
\[
30p + 30r = 30(12) + 30(5)
\]
\[
= 360 + 150
\]
\[
= 510
\]

### Answer
The cost of 30 pens and 30 rulers is **$510.00**.

Would you like more details, or do you have any questions?

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Here are some related questions that build on this topic:

1. What would be the total cost if we bought 10 pens and 10 rulers?
2. How would the answer change if the cost of each pen increased by $2?
3. Can you solve for different quantities of pens and rulers using the same method?
4. What is the minimum number of pens and rulers we need to buy to spend exactly $100?
5. How can we use matrix methods to solve this type of problem?

**Tip:** When solving linear systems, it's often helpful to simplify equations first before substituting or using elimination. This reduces potential calculation errors.

If 8 pens and 4 rulers cost $116.00 and 2 pens and 6 rulers cost $54.00, find the cost of 30 pens and 30 rulers.

Learn how to determine the cost of 30 pens and 30 rulers by solving a system of linear equations. This problem covers key algebraic concepts, including substitution and elimination methods. Suitable for students in Grades 8-10.

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