https://file.aiquickdraw.com/mathbot/file/1726553632587fxk6u4mb.jpg

The problem involves three items: pencils, pens, and erasers, with two sets of data provided, and you need to find the cost for a new set of quantities. Let's break it down:

Given:
1. \( 5 \text{ pencils} + 9 \text{ pens} + 13 \text{ erasers} = 190 \)
2. \( 9 \text{ pencils} + 5 \text{ pens} + 19 \text{ erasers} = 190 \)

You are asked to find the total cost of:
- \( 30 \text{ pencils} + 26 \text{ pens} + 67 \text{ erasers} \)

Let's represent the unknown costs as follows:
- Let the cost of one pencil be \( x \)
- Let the cost of one pen be \( y \)
- Let the cost of one eraser be \( z \)

This gives us the system of equations:
1. \( 5x + 9y + 13z = 190 \)
2. \( 9x + 5y + 19z = 190 \)

We need to solve this system to find the values of \( x \), \( y \), and \( z \), and then use those values to calculate the total cost of \( 30x + 26y + 67z \).

Let me solve this step-by-step.The solution to the system of equations is expressed in terms of \( z \) (the cost of one eraser):

\[
x = \frac{95}{7} - \frac{53z}{28}
\]
\[
y = \frac{95}{7} - \frac{11z}{28}
\]

Since we have a free variable \( z \), we can't determine unique values for the costs of pencils and pens without more information. However, we can express the total cost for 30 pencils, 26 pens, and 67 erasers using this result.

Let's calculate the total cost \( 30x + 26y + 67z \) in terms of \( z \).It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like me to walk through the steps of simplifying this expression manually or clarify any details so far?

Here are some related questions for further exploration:
1. How can the cost of pencils, pens, and erasers be expressed using other variables?
2. What strategies are there to solve systems of equations with multiple variables?
3. How would the solution change if another piece of data were provided?
4. Can these kinds of equations be solved graphically or by other methods?
5. How would you approach solving this system if only estimates were allowed?

**Tip:** Always check if more information can reduce the number of variables before trying to solve complex systems of equations.

Find the cost of 30 pencils, 26 pens, and 67 erasers given the following information: 5 pencils + 9 pens + 13 erasers = 190 and 9 pencils + 5 pens + 19 erasers = 190.

Solve for the cost of 30 pencils, 26 pens, and 67 erasers using a system of linear equations. This problem demonstrates algebraic concepts, including solving simultaneous equations through the substitution or elimination methods. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation