In each diagram shown in this problem, the number on the line connecting two circles is the sum of the two numbers in these two circles. An example of a completed diagram is shown to the right.

![](https://cdn.mathpix.com/cropped/2024_04_17_ab5f9723a8aff65a55ebg-3.jpg?height=268&width=308&top_left_y=511&top_left_x=1428)

(a) What is the value of $x$ ?

![](https://cdn.mathpix.com/cropped/2024_04_17_ab5f9723a8aff65a55ebg-3.jpg?height=262&width=309&top_left_y=758&top_left_x=997)

(b) With justification, determine the value of $y$.
![](https://cdn.mathpix.com/cropped/2024_04_17_ab5f9723a8aff65a55ebg-3.jpg?height=594&width=328&top_left_y=1076&top_left_x=992)

(c) With justification, determine the values of $p, q$ and $r$.
Through this solution we call the number in the top left circle the "left number", the number in the top right circle the "right number", and the number in the bottom circle the "bottom number".

Further, we call the line joining the left and right numbers the "top line", the line joining the left and bottom numbers the "left line" and the line joining the right and bottom numbers the "right line".

(a) Since the sum of the numbers along the top line is 13 , then the left number is $13-4=9$.

Since the sum of the numbers along the left line is 10 , then the bottom number is $10-9=1$.

Since $x$ is the sum of the numbers along the right line, then $x=1+4=5$.

![](https://cdn.mathpix.com/cropped/2024_04_17_6308d40ff1469d7b994ag-06.jpg?height=265&width=312&top_left_y=621&top_left_x=1579)

(b) From the left line, $3 w+w=y$ or $y=4 w$.

Since the sum of the numbers along the right line is also $y$, then the right number is $y-w=4 w-w=3 w$.

From the top line, $3 w+3 w=48$ or $6 w=48$ and so $w=8$.

Therefore, $y=4 w=32$.

![](https://cdn.mathpix.com/cropped/2024_04_17_6308d40ff1469d7b994ag-06.jpg?height=271&width=312&top_left_y=927&top_left_x=1579)

(c) From the top line, $p+r=3$.

From the left line, $p+q=18$.

From the right line, $q+r=13$.

There are many different ways of solving this system of three equations in three unknowns. Here are two methods of doing this.

![](https://cdn.mathpix.com/cropped/2024_04_17_6308d40ff1469d7b994ag-06.jpg?height=263&width=312&top_left_y=1240&top_left_x=1579)

Method 1

We add the three equations to obtain $(p+r)+(p+q)+(q+r)=3+18+13$ or $2 p+2 q+2 r=34$.

Dividing by 2 , we obtain $p+q+r=17$.

Therefore, $p=(p+q+r)-(q+r)=17-13=4$ and $q=(p+q+r)-(p+r)=17-3=14$

and $r=(p+q+r)-(p+q)=17-18=-1$.

(We can check these values in the original diagram.)

Method 2

From the first equation $r=3-p$.

Substituting this into the third equation, we obtain $q+(3-p)=13$ or $q-p=10$.

Therefore, we have $p+q=18$ and $q-p=10$.

Adding these two equations, we obtain $2 q=28$ or $q=14$ and so $p=4$, whence $r=-1$.

There are many other ways to solve this system of equations.

The required values are $p=4, q=14$ and $r=-1$.