question in description!!!

Question

When you reverse the digits in a certain two-digit number, you decrease its value by 54. The sum of its digits is 10.

Write a system of equations that models this scenario.

 Using the table, determine the number described

Answer 1

Let the two-digit number be n, with digits xy.

 

n = 10x + (1)y = 10x + y

 

Given:  x + y = 10

 

If you reverse the digits, you get a new number 

 

m = 10y + x = n - 54 = (10x + y) - 54

 

⇒ 10y + x = (10x + y) - 54 ⇒ -9x + 9y = -54

 

Set up and solve a system of two equations in two unknowns:

 

x + y = 10

-9x + 9y = -54

 

Simplify and solve:

 

x + y = 10

-x + y = -6

 

2y = 4

 

y = 2

 

x = 10 - y = 10 - 2 = 8

 

The original two digit number is 82.  The number with the digits reversed is 28.

 

Test the solution:  8 + 2 = 2 + 8 = 10 [good]

 

82 - 54 = 28 [good]

 

 

Answer 2

Tiara,

 

d1 = digit 1 (ones' position)

d2 = digit 2 (tens' position)

 

 

First constraint:

d2 + d1 = 10

 

Second constraint:

d1 * 10 + d2 = 10 * d2 + d1 - 54 (represent the 2-digit number in terms of its power-of-ten position)

 

 

Simplify the second constraint:

9 * d1 - 9 * d2 = - 54

d1 - d2 = - 6

 

Rewrite the system of equations using the simplified second equation:

d1 + d2 = 10

d1 - d2 = - 6

 

Solve the above system of equations by choosing ONE variable (the chosen variable does not matter):

d1 = 10 - d2

 

Substitute the above equation into the second equation of the system:

(10 - d2) - d2 = - 6

10 - 2 * d2 = - 6

-2 * d2 = - 16

d2 = 8

 

Choose either of the equations in the above system to get the ones' digit (d1):

d1 + d2 = 10

d1 + 8 = 10

d1 = 2

 

Ans: d1 = 2 (ones' position); d2 = 8 (tens' position)

 

Original 2-digit number: 82

Reversed 2-digit number: 28

Difference between the Original and Reversed numbers: 82 - 28 = 54.

 

I hope this helps,

 

Gary

Answer 3

Let the first number = x and the second number = y. We know that x + y = 10.

 

We also know that 10x + y - (10y + x) = 54

 

or 10x + y - 10y - x = 54

 

9x - 9y = 54

 

x - y = 6

 

We have the system 

 

x + y = 10

x - y = 6

 

Solve the second equation for x.

 

x = y + 6

 

Now, plug this into the first equation 

 

(y + 6) + y = 10

 

2y + 6 = 10

 

2y = 4

 

y = 2

 

If y = 2, then x = y + 6 = 2 + 6 = 8

 

The number is 82 because 82 - 28 = 54