the sum of three numbers is 8. the first number plus the second is 16 less than the third. the second number is 8 more than the third. find the numbers
a freight train leaves detroit traveling east at a speed of 65mph. one hour later, a passanger train leaves traveling wst a 85mph. how long will it take for the trains to have traveled the sam distance?
a + b + c = 8
a + b = c - 16
b = c + 8
so
c - 16 + c = 8
2 c = 24
c = 12
b = c + 8 = 20
a + 20 + 12 = 8
a + 32 = 8
a = -24
To solve this problem, we'll use algebraic equations.
Let's represent the three unknown numbers as variables:
Let the first number be x.
Let the second number be y.
Let the third number be z.
Now we can create equations based on the given information:
Equation 1: x + y + z = 8 (The sum of the three numbers is 8.)
Equation 2: x + y = z - 16 (The first number plus the second number is 16 less than the third.)
Equation 3: y = z + 8 (The second number is 8 more than the third.)
Now we have a system of three equations. We can solve this system by substitution or elimination method. Let's use substitution:
From Equation 3, we can rewrite it as z = y - 8.
Substitute z = y - 8 into Equation 2:
x + y = (y - 8) - 16
x + y = y - 24 (distribute the -16)
x = -24 (Subtract y from both sides)
Now we have the value of x.
Substitute x = -24 into Equation 1:
-24 + y + z = 8
y + z = 32 (Add 24 to both sides)
Now we have an equation with y and z. We can substitute y = z + 8 into this equation:
(z + 8) + z = 32
2z + 8 = 32 (Combine like terms)
2z = 24 (Subtract 8 from both sides)
z = 12 (Divide both sides by 2)
We have found the value of z.
Now, substitute z = 12 into Equation 3:
y = 12 + 8
y = 20
Lastly, substitute y = 20 and z = 12 into Equation 1:
x + 20 + 12 = 8
x + 32 = 8 (Combine like terms)
x = -24
The three numbers are:
- x = -24
- y = 20
- z = 12