The sum of 3 numbers is 24. Twice the smallest number is 2 less than the largest number and the largest number is equal to the sum of the other two. what are the three numbers?
x+y+z = 24
2x = z-2
z = x+y
Now just solve for x,y,z
To solve this problem, let's start by assigning variables to each of the numbers. Since we don't know the values of the numbers, we can represent them as x, y, and z.
According to the problem statement, we have three equations:
1) x + y + z = 24 (The sum of the three numbers is 24)
2) 2x = z - 2 (Twice the smallest number is 2 less than the largest number)
3) z = x + y (The largest number is equal to the sum of the other two)
Now, let's solve the system of equations step by step:
From equation 3, we can replace z with (x + y) in equation 2:
2x = (x + y) - 2
Expanding the equation:
2x = x + y - 2
By moving the x term to the other side, we get:
x = y - 2
Now, substitute this value of x into equation 1:
(y - 2) + y + z = 24
Simplifying the equation:
2y + z = 26
Now, substitute the value of z from equation 3 (z = x + y) into the equation above:
2y + (x + y) = 26
Expanding the equation:
2y + x + y = 26
Combining like terms, we have:
3y + x = 26
Now, we have a system of two equations:
x = y - 2
3y + x = 26
To solve this system, we can use substitution or elimination method. Let's solve it using the substitution method:
Rearrange the equation x = y - 2 to x - y = -2:
x - y = -2
Now, substitute this value of x into the other equation:
3y + (y - 2) = 26
Simplifying the equation:
4y - 2 = 26
Add 2 to both sides:
4y = 28
Divide both sides by 4:
y = 7
Now, substitute the value of y into equation 1:
x + 7 + z = 24
Simplifying the equation:
x + z = 17
Since z = x + y, substitute the value of y into this equation:
z = x + 7
Now, substitute the value of z in the equation above into x + z = 17:
x + (x + 7) = 17
Simplifying the equation:
2x + 7 = 17
Subtract 7 from both sides:
2x = 10
Divide both sides by 2:
x = 5
Finally, substitute the values of x and y into the equation z = x + y:
z = 5 + 7
z = 12
Therefore, the three numbers are 5, 7, and 12.