A two digit number is such that two times the tens digit is three less than thrice the units digit and four times the number is 99 greater than the number obtained by reversing the digits. Find the number.
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To solve this problem, we need to analyze the given information and set up equations based on the problem statement.
Let's assume that the tens digit of the two-digit number is 'x' and the units digit is 'y'. Based on the problem statement, we can write the following equations:
Equation 1: "Two times the tens digit is three less than thrice the units digit."
This can be represented as:
2x = 3y - 3
Equation 2: "Four times the number is 99 greater than the number obtained by reversing the digits."
This can be represented as:
4(10x + y) = (10y + x) + 99
Now, let's solve the equations to find the values of x and y.
Simplifying Equation 1:
2x = 3y - 3
Rearranging terms, we get:
2x - 3y = -3 --(Equation 3)
Simplifying Equation 2:
4(10x + y) = (10y + x) + 99
Expanding the multiplication, we get:
40x + 4y = 10y + x + 99
Combining like terms, we get:
40x + 4y - 10y - x = 99
Simplifying further:
39x - 6y = 99 --(Equation 4)
Now, we have two equations: Equation 3 and Equation 4. We can solve this system of equations by either substitution or elimination method.
Let's use the elimination method:
Multiply Equation 3 by 39 and Equation 4 by 2 to make the coefficients of 'x' equal:
78x - 117y = -117 --(Equation 5)
78x - 12y = 198 --(Equation 6)
Now, subtract Equation 6 from Equation 5 to eliminate 'x':
(78x - 117y) - (78x - 12y) = -117 - 198
Simplifying:
78x - 117y - 78x + 12y = -315
Combining like terms:
-105y = -315
Simplifying further:
y = -315 / -105
y = 3
Substitute the value of 'y' back into Equation 3 to solve for 'x':
2x - 3(3) = -3
2x - 9 = -3
Adding 9 to both sides:
2x = 6
Dividing by 2:
x = 3
Therefore, the tens digit 'x' is 3 and the units digit 'y' is 3. Thus, the two-digit number is 33.