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Find the numbers to place in each of the five small circles shown to the right so that each number given in a large circle is the sum of the numbers in the two adjacent small circles. Do so by letting one of the values in a small circle be denoted by an unknown and then obtaining and solving an equation for the unknown. In clockwise order from the​ bottom, the entries in the small circles are 



To solve this problem, we can denote the unknown values in the small circles as variables. Let's label the five small circles (clockwise from the bottom) as \( x_1, x_2, x_3, x_4, x_5 \).

The large circles provide the sum of two adjacent small circles. Based on this, we have the following system of equations:

1. \( x_1 + x_2 = 18 \) (for the bottom large circle)
2. \( x_2 + x_3 = 28 \) (for the left large circle)
3. \( x_3 + x_4 = 34 \) (for the top large circle)
4. \( x_4 + x_5 = 16 \) (for the right large circle)
5. \( x_5 + x_1 = 26 \) (for the bottom right large circle)

Now, I will solve this system of equations by substitution or elimination method to find the values of \( x_1, x_2, x_3, x_4, x_5 \).

Let me compute that for you.The values of the numbers in the small circles, in clockwise order from the bottom, are:

- \( x_1 = 17 \)
- \( x_2 = 1 \)
- \( x_3 = 27 \)
- \( x_4 = 7 \)
- \( x_5 = 9 \)

Would you like further details or have any other questions?

Here are five related questions to further explore this topic:

1. How do we check the consistency of the solution with the original problem?
2. How can we generalize this system to a larger number of circles?
3. What is the process to solve such a system using matrix methods?
4. How does substitution compare with elimination when solving systems of linear equations?
5. How does the number of equations relate to the number of unknowns in such problems?

**Tip:** When solving systems of equations, always double-check your final answers by substituting them back into the original equations.

Find the numbers to place in each of the five small circles shown to the right so that each number given in a large circle is the sum of the numbers in the two adjacent small circles. Do so by letting one of the values in a small circle be denoted by an unknown and then obtaining and solving an equation for the unknown. In clockwise order from the​ bottom, the entries in the small circles are...

This problem involves finding unknown values in a system of linear equations based on a circle puzzle where the sum of numbers in adjacent small circles equals the number in the large circle. A step-by-step solution is provided for students in Grades 9-11.

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