a two digit number is four times the sum of its digits and twice the product of the digit. find the number using elimination method
let the tens digit be x
let the unit digit be y
then the number is 10x+y
10x+y = 4(x+y) ----> 6x - 3y = 0
6x = 3y
y = 2x
10x + y = 2(xy)
10x - 2xy + y = 0
10x - 2x(2x) + 2x = 0
-4x^2 + 12x = 0
x^2 - 3x= 0
x(x-3) = 0
x = 0 or x = 3 , but if x=0 you don't have a 2-digit number
x = 3, y = 6
the number is 36
I used substitution instead of elimination since it was clearly the easiest and most obvious way to do it, because of the simple result of x=2x from the first equation.
the xy term of the second equation would lead to a more complicated result using elimination.
To find the two-digit number using the elimination method, we need to set up a system of equations based on the given information.
Let's assume that the tens digit of the number is x and the ones digit is y.
According to the problem, the number is four times the sum of its digits and twice the product of its digits. We can express this information in the form of equations:
Equation 1: 10x + y = 4(x + y)
Equation 2: 10x + y = 2xy
To solve this system of equations using the elimination method, we can subtract Equation 1 from Equation 2 to eliminate the variable y:
(10x + y) - (10x + y) = 2xy - 4(x + y)
0 = 2xy - 4x - 4y
Now, we can factor out 2 from the right side of the equation:
0 = 2(x - 2)(y - 2)
Since the product of two numbers is zero, one of these factors must be zero:
Case 1: x - 2 = 0
x = 2
Case 2: y - 2 = 0
y = 2
Plugging these values back into either Equation 1 or Equation 2 will give us the complete solution.
Let's choose Equation 1:
10x + y = 4(x + y)
Substituting x = 2, we have:
10(2) + y = 4(2 + y)
20 + y = 8 + 4y
Rearranging the equation:
3y = 12
y = 4
Therefore, the two-digit number is 24.