he sum of three integers is 193. The sum of the first and second integers exceeds the third by 85. The third integer is 15 less than the first. Find the three integers.
Do they have to be consecutive integers?
One integer will be 54, the other two could be any positive numbers whose average is 69.5
69 and 70, or 68 and 71, or 67 and 72...
To solve this problem, let's assign variables to the three integers.
Let's call the first integer "x."
Let's call the second integer "y."
And let's call the third integer "z."
According to the problem, we have three conditions to work with:
1. The sum of the first, second, and third integers is 193:
x + y + z = 193
2. The sum of the first and second integers exceeds the third by 85:
x + y = z + 85
3. The third integer is 15 less than the first:
z = x - 15
Now, we have three equations, and we can use them to solve for the values of x, y, and z.
First, let's substitute equation 3 into equation 2:
x + y = (x - 15) + 85
Next, let's simplify this equation:
x + y = x + 70
Subtracting x from both sides gives us:
y = 70
Now let's substitute the value of y into equations 1 and 2:
1. x + 70 + z = 193
2. x + 70 = z + 85
From equation 2, we can solve for z by rearranging the equation:
z = (x + 70) - 85
z = x - 15
Since both equations 1 and 2 equal z, we can set them equal to each other:
x + 70 + z = x - 15
Next, let's solve for x:
Simplifying this equation gives us:
70 + z = -15
Subtracting z from both sides gives us:
70 = -15 - z
Adding 15 to both sides gives us:
85 = -z
Finally, multiplying both sides by -1 gives us:
z = -85
Now that we have the value of z, let's substitute it back into equation 1:
x + 70 + (-85) = 193
Simplifying this equation gives us:
x - 15 = 193
Adding 15 to both sides gives us:
x = 208
Now, we have the values of x and z. We can substitute them back into equation 2 to find y:
x + 70 = z + 85
208 + 70 = -85 + 85
278 = 0
Since the equation is not true, it means there is no solution for y.
Therefore, the three integers that satisfy all the conditions are:
x = 208
y = No solution
z = -85