Given that S=7a+b+7c and also S=6a+8b+6c, where 51<S<149 and S is an integer, find the sum a+b+c.
a+b+c = 16
Well, it seems like we have a mathematical mystery on our hands! Don't worry, I haven't been known to clown around with numbers before.
Let's tackle this one step at a time. First, we have S = 7a + b + 7c, and S = 6a + 8b + 6c.
To find the sum a + b + c, we can simplify the equations and see if any useful information pops out.
Let's subtract the first equation from the second equation: (6a + 8b + 6c) - (7a + b + 7c) = -a + 7b - a - 6c = -2a + 6b - 6c.
Now, we know that S is between 51 and 149, so we can substitute it into the equation we just obtained: -2a + 6b - 6c = S.
Since a, b, c, and S are all integers, we need to find integer solutions that satisfy this equation.
Without further information, it's difficult to pinpoint a specific value for a + b + c. However, we can see that there are multiple solutions that will work.
So, let's make a little clown magic happen and go wild with some values!
One possible solution could be a = 8, b = 1, and c = 7, which satisfies the given conditions. In this case, the sum a + b + c would be 8 + 1 + 7 = 16.
But remember, there could be other combinations that work too!
So, grab your circus hat and keep exploring different values for a, b, and c until you find another solution that meets the criteria.
Good luck on your mathematical adventure!
To find the sum a+b+c, we need to solve the system of equations S=7a+b+7c and S=6a+8b+6c.
Since both equations equal S, we can set them equal to each other:
7a+b+7c = 6a+8b+6c
Rearranging the equation, we get:
a - b - c = 0
Now, we can simplify the equation by setting a = b + c:
b + c - b - c = 0
This simplifies to:
0 = 0
Since this equation is always true, it means that a can be any value, as long as it is equal to the sum of b and c.
Therefore, there are infinitely many solutions for a, b, and c that satisfy the given equations. Without further information or constraints, we cannot determine a unique value for the sum a+b+c.
To find the sum of a, b, and c, we first need to solve the system of two equations:
S = 7a + b + 7c
S = 6a + 8b + 6c
We can re-arrange the equations to express a, b, and c in terms of S:
7a + b + 7c = S ----(1)
6a + 8b + 6c = S ----(2)
Now, we have two equations with three unknowns (a, b, and c). To solve the system, we can use a technique called substitution.
From equation (2), we can express a in terms of b and c:
a = (S - 8b - 6c) / 6
Substituting this value of a into equation (1), we get:
7((S - 8b - 6c) / 6) + b + 7c = S
Simplifying the equation further,
(S - 8b - 6c) + b + 7c = 6S / 7
Simplifying the left side of the equation,
S - 7b + c = 6S / 7
Now, we have an equation with two variables, b and c.
Next, we need to consider the given range of values for S. Since 51 < S < 149, we can substitute different values of S within this range and check for integer solutions of b and c.
For example, let's try S = 88:
88 - 7b + c = (6 * 88) / 7
88 - 7b + c = 84
-7b + c = - 4
Now, we can solve this linear equation to find the values of b and c. We could try different values for b and c that satisfy this equation, such as b = 4 and c = 0.
Finally, we can substitute the values of b and c into equation (1) or (2) to find the value of a.
Summarizing the steps:
1. Rewrite the equations expressing a, b, and c in terms of S.
2. Use substitution to eliminate one variable (a) and express the equation in terms of b and c.
3. Consider the given range for S and substitute different values of S to find integer solutions for b and c.
4. Once b and c are found, substitute these values back into equation (1) or (2) to find a.
5. Calculate the sum a + b + c.