solve positive integers a, b, and c so that 33a+5=20b+15=29c+17
To solve the system of equations:
33a + 5 = 20b + 15 = 29c + 17
We need to understand that each equation represents a relationship between the variables a, b, and c.
Let's start by finding a relationship between two of the equations:
33a + 5 = 20b + 15
We can simplify this equation by isolating one variable in terms of the other variable. Let's isolate a in terms of b:
33a = 20b + 10
a = (20b + 10) / 33
Now we have a relationship between a and b. Let's substitute this equation into the third equation:
29c + 17 = (20b + 10) / 33
To simplify this equation and isolate c, we need to get rid of the fraction. Multiply both sides of the equation by 33:
33(29c + 17) = 20b + 10
969c + 561 = 20b + 10
Now, let's simplify the equation further:
969c = 20b - 551
c = (20b - 551) / 969
Now we have a relationship between c and b. We can substitute this equation back into the first equation:
33a + 5 = (20b + 10) / 33
To simplify this equation and isolate a, we multiply both sides by 33:
33(33a + 5) = 20b + 10
1089a + 165 = 20b + 10
Rearranging the equation:
1089a = 20b - 155
a = (20b - 155) / 1089
Now we have a relationship between a and b. We can substitute this equation back into the second equation:
20b + 15 = 29c + 17
To simplify this equation, we isolate b:
20b = 29c + 2
b = (29c + 2) / 20
Now we have expressions for all three variables a, b, and c in terms of each other:
a = (20b + 10) / 33
b = (29c + 2) / 20
c = (20b - 551) / 969
To find the values for a, b, and c, we can start by choosing a value for one of the variables and substituting it into the equations to solve for the other variables.
Since we are looking for positive integers, we can start with b = 1 and solve for a and c. Now we substitute b = 1 into the equations:
a = (20 * 1 + 10) / 33 = 30/33 = 10/11
c = (20 * 1 - 551) / 969 = -531/969
Since we want positive integers, these values do not satisfy our conditions.
We can try other values of b until we find positive integer solutions for a, b, and c.