Allison, Ashley, and Becky weigh 124 kg altogether. Allison and Becky weigh 102 kg altogether. Allison and Ashley weigh 86 kg altogether. How much does each person weigh? please show your work.
Let's solve this problem step by step using a system of equations.
Step 1: Assign variables to the unknown weights.
- Let's say the weight of Allison is A kg.
- Let's say the weight of Ashley is B kg.
- Let's say the weight of Becky is C kg.
Step 2: Translate the given information into equations.
- "Allison, Ashley, and Becky weigh 124 kg altogether" can be expressed as:
A + B + C = 124 Equation 1
- "Allison and Becky weigh 102 kg altogether" can be expressed as:
A + C = 102 Equation 2
- "Allison and Ashley weigh 86 kg altogether" can be expressed as:
A + B = 86 Equation 3
Step 3: Solve the equations.
We can solve this system of equations using substitution or elimination method. Let's use the elimination method here:
To eliminate variable A, subtract Equation 2 from Equation 3:
(A + B) - (A + C) = 86 - 102
B - C = -16 Equation 4
Now, we have two equations:
B - C = -16 Equation 4
A + B + C = 124 Equation 1
To eliminate variable B, add Equation 4 to Equation 1:
(A + B + C) + (B - C) = 124 + (-16)
A + 2B = 108 Equation 5
Now, we have two equations:
A + 2B = 108 Equation 5
B - C = -16 Equation 4
Step 4: Solve the remaining equations.
To solve this system of linear equations, we can substitute the value of B from Equation 4 into Equation 5. This will give us the value of A. We can then substitute the value of A into Equation 2 to find the value of C.
Substituting B = -16 + C in Equation 5:
A + 2(-16 + C) = 108
A - 32 + 2C = 108
A + 2C = 140 Equation 6
Substituting A = 140 - 2C in Equation 2:
(140 - 2C) + C = 102
140 - C = 102
C = 140 - 102
C = 38
Substituting C = 38 in Equation 4:
B - 38 = -16
B = -16 + 38
B = 22
Finally, substituting B = 22 and C = 38 in Equation 1:
A + 22 + 38 = 124
A + 60 = 124
A = 124 - 60
A = 64
So, Allison weighs 64 kg, Ashley weighs 22 kg, and Becky weighs 38 kg.