The sum of 3 integers is 194. the sum of the first and second integers exceeds the third by 80. The third integer is 45 less than the first. Find the three integers
let the first be x
let the second be y
the third is x-45
x+y > x-45 by 80 ---> x+y = x-45 + 80
x + y = x + 35
y = 35
x + y + x-45 = 194
2x + 35 - 45 = 194
2x = 204
x = 102
the first is 102
the 2nd is 35
the 3rd is 57
Well, let's call our first integer "x", our second integer "y", and our third integer "z".
From the information given, we can set up a system of equations:
Equation 1: x + y + z = 194
Equation 2: x + y = z + 80
Equation 3: z = x - 45
Now, let's solve this puzzle with a little bit of clown magic!
Equation 2 in terms of z: x + y = (x - 45) + 80
Simplifying, x + y = x + 35
Subtracting x from both sides, we get: y = 35
Now, we substitute this value of y into Equation 1: x + 35 + z = 194
Substituting Equation 3 (z = x - 45), we have: x + 35 + (x - 45) = 194
Simplifying, 2x - 10 = 194
Adding 10 to both sides, we get: 2x = 204
Dividing by 2: x = 102
Using Equation 3, we find z: z = 102 - 45
Thus, z = 57
Now we can check our work using Equation 1: 102 + 35 + 57 = 194
Therefore, the three integers are 102, 35, and 57.
Let's represent the three integers as x, y, and z.
From the given information, we have the following equations:
1) x + y + z = 194 (sum of the three integers)
2) x + y = z + 80 (sum of the first and second integers exceeds the third by 80)
3) z = x - 45 (third integer is 45 less than the first)
We can solve these equations step by step to find the values of x, y, and z.
Step 1: Substituting equation 3 into equation 2
x + y = (x - 45) + 80
x + y = x + 35 (simplifying the equation)
y = 35
Step 2: Substituting y = 35 into equations 1 and 2
x + 35 + z = 194 (equation 1)
x + 35 = z + 80 (equation 2)
Step 3: Substituting z + 80 for x + 35 in equation 1
(z + 80) + 35 + z = 194
2z + 115 = 194 (simplifying the equation)
Step 4: Solving for z in equation 3
2z = 194 - 115
2z = 79
z = 79 / 2
z = 39.5
Step 5: Substituting z = 39.5 into equation 2
x + 35 = 39.5 + 80
x + 35 = 119.5
x = 119.5 - 35
x = 84.5
Therefore, the three integers are approximately 84.5, 35, and 39.5.
To solve this problem, we can set up a system of equations to represent the given information.
Let's assume the three integers as:
First integer: x
Second integer: y
Third integer: z
From the problem statement, we have three equations:
1) x + y + z = 194 (sum of the three integers is 194)
2) x + y = z + 80 (sum of the first and second integers exceeds the third by 80)
3) z = x - 45 (the third integer is 45 less than the first)
Now, we can solve this system of equations using substitution or elimination method.
Let's use substitution method:
Substitute equation 3 into equation 2:
x + y = (x - 45) + 80
Simplify:
x + y = x + 35
Subtract x from both sides:
y = 35
Now, substitute this value of y into equation 1:
x + 35 + z = 194
Subtract 35 from both sides:
x + z = 159
Since we have two equations with two variables (x and z), we can solve for them.
Now, subtract equation 3 from equation 1:
(x + z) - (x - 45) = 159 - 45
Simplify:
z + 45 = 114
Subtract 45 from both sides:
z = 69
Substitute this value of z back into equation 1:
x + 35 + 69 = 194
Combine like terms:
x + 104 = 194
Subtract 104 from both sides:
x = 90
So, the three integers are:
First integer: x = 90
Second integer: y = 35
Third integer: z = 69
Therefore, the three integers are 90, 35, and 69.