Alice, Bob, and Charlie are three siblings with ages a, b, and c respectively. Their combine age is 18 years. Charlie is older than Bob. Alice's age is equal to twice the difference between Bob's and Charlie's ages. Bob's age is equal the average of Alice's and Charlie's ages.
(a)Use the information provided to formulate a system of three equations for a, b, and c.
(b)Solve the system using row reduction techniques, or otherwise.
What's the trouble? Just start writing the words as equations. We are told:
a+b+c = 18
a = 2(c-b) (because c > b)
b = (a+c)/2
Now just go through the usual procedures to solve several equations.
i dunno so stop
To solve this problem, let's start by assigning variables to the ages of Alice, Bob, and Charlie.
Let a represent Alice's age, b represent Bob's age, and c represent Charlie's age.
a) Formulating a system of equations:
1. The combined age of Alice, Bob, and Charlie is 18 years: a + b + c = 18.
2. Charlie is older than Bob: c > b.
3. Alice's age is twice the difference between Bob's and Charlie's ages: a = 2(c - b).
4. Bob's age is equal to the average of Alice's and Charlie's ages: b = (a + c) / 2.
Now we have a system of three equations:
a + b + c = 18 (Equation 1)
c > b (Equation 2)
a = 2(c - b) (Equation 3)
b = (a + c) / 2 (Equation 4)
b) Solving the system:
To solve the system of equations, we can use row reduction techniques or substitute the variables to eliminate them gradually.
Substituting Equation 4 into Equation 1:
a + ((a + c) / 2) + c = 18
Multiplying the equation by 2 to eliminate the fraction:
2a + a + c + 2c = 36
3a + 3c = 36
Dividing by 3:
a + c = 12 (Equation 5)
Substituting Equation 3 into Equation 4:
(2(c - b)) = (a + c) / 2
Multiplying both sides by 2 to eliminate the fraction:
4(c - b) = a + c
4c - 4b = a + c
4c - a - c = 4b
3c - a = 4b (Equation 6)
Now that we have two equations, Equation 5 and Equation 6, we can solve for a, b, and c.
Using Equation 5, we solve for a:
a = 12 - c (Equation 7)
Substituting Equation 7 into Equation 6:
3c - (12 - c) = 4b
4c - 12 = 4b
4c - 4b = 12
c - b = 3 (Equation 8)
From Equation 8, we can conclude that c = b + 3.
Now we have three variables, a, b, and c, in terms of each other:
c = b + 3 (Equation 9)
a = 12 - c (Equation 10)
To solve for a, b, and c, we need to make further assumptions or gather more information about their specific ages. Without additional information, we cannot find unique values for a, b, and c.