find three numbers such that the second is 6 less than 3 times the first , and the third is 2 more than 2/3 of the 2nd.The sum of the three numbers is 172.
use x=1st number
y=second number
z= 3rd number
please help...thanks a lot!!
You have 3 consecutive questions of this type.
Why do you want to use 3 variables, it is much easier to just define one, and the use the given to define the others.
e.g.
let the first number be x
then the 2nd number is 3x-6
and the 3rd is (2/3)(3x-6) + 2 = 2x -4 + 2 = 2x - 2
then x + 3x-6 + 2x-2 = 172
6x = 180
x = 30
1st = 30
2nd = 3(30)-6 = 84
3rd = 58
check: 30+84+58 = 172
try the others using only one variable.
Let's break down the problem step by step:
Step 1: Assign variables
Let's assign variables to the unknown numbers:
x = first number
y = second number
z = third number
Step 2: Translate the first condition into an equation
The second number is 6 less than 3 times the first number. This can be expressed as:
y = 3x - 6
Step 3: Translate the second condition into an equation
The third number is 2 more than 2/3 of the second number. This can be expressed as:
z = (2/3)y + 2
Step 4: Translate the third condition into an equation
The sum of the three numbers is 172. This can be expressed as:
x + y + z = 172
Now we have a system of three equations with three variables. We can solve this system to find the values of x, y, and z.
Step 5: Solve the system of equations
Let's start by substituting the value of y from equation 2 into equations 1 and 3:
x + (3x - 6) + ((2/3)(3x - 6) + 2) = 172
Now, simplify and solve for x:
x + 3x - 6 + (2/3)(3x - 6) + 2 = 172
Multiply (2/3) by (3x - 6):
x + 3x - 6 + (2/3)(3x) - (2/3)(6) + 2 = 172
Simplify further:
x + 3x - 6 + 2x - 4 + 2 = 172
6x - 8 = 172
Add 8 to both sides:
6x = 180
Divide both sides by 6:
x = 30
Now that we have the value of x, we can substitute it back into equation 2 to solve for y:
y = 3x - 6
y = 3(30) - 6
y = 90 - 6
y = 84
Finally, substitute the values of x and y into equation 3 to solve for z:
z = (2/3)y + 2
z = (2/3)(84) + 2
z = 56 + 2
z = 58
Therefore, the three numbers are:
x = 30
y = 84
z = 58
To solve this problem, we can set up a system of three equations based on the given information.
Let's translate the given statements into equations:
1. "The second number is 6 less than 3 times the first number."
y = 3x - 6
2. "The third number is 2 more than two-thirds of the second number."
z = (2/3)y + 2
3. "The sum of the three numbers is 172."
x + y + z = 172
Now we have a system of three equations:
Equation 1: y = 3x - 6
Equation 2: z = (2/3)y + 2
Equation 3: x + y + z = 172
To solve this system of equations, we can substitute the values from Equation 1 and Equation 2 into Equation 3:
x + (3x - 6) + ((2/3)(3x - 6) + 2) = 172
Simplifying the equation:
x + 3x - 6 + (2x - 4) + 2 = 172
6x - 8 = 172
Adding 8 to both sides:
6x = 180
Dividing both sides by 6:
x = 30
Now that we have the value of x, we can substitute it back into Equation 1 to find y:
y = 3(30) - 6
y = 90 - 6
y = 84
Finally, we can substitute the values of x and y into Equation 2 to find z:
z = (2/3)(84) + 2
z = 56 + 2
z = 58
So the three numbers are:
First number (x) = 30
Second number (y) = 84
Third number (z) = 58