A number consists of two digits. The number is 2 more than 8 times the sum of the digits, and if 54 is subtracted from the number, the digits will be reversed. Find the number.

Let the units digit = x

Let the 10s digit = y

Equations

10y + x - 2 = 8 * (x + y) (1)

10y + x - 54 = 10x + y (2)

Equation 1

10y + x - 2 = 8 * (x + y) Remove the brackets on the right

10y + x - 2 = 8x + 8y Subtract 8y from both sides

10y - 8y + x - 2 = 8x Subtract 8x from both sides

2y + x - 8x - 2 = 0 Combine

2y - 7x - 2 = 0 (3)

Equation 2

10y + x - 54 = 10x + y Subtract 10x

10y - 9x - 54 = y Subtract y from both sides.

9y - 9x - 54 = 0 Divide by 9

y - x - 6 = 0 Multiply by 2

2y - 2x - 12 = 0 (4)

Solution

Subtract (3) - (4)

2y - 7x - 2 = 0

2y - 2x - 12 = 0

- 5x + 10 = 0 Subtract 10 from both sides.

- 5x = - 10 Divide by - 5

x = 2

Use (4) to find y

2y - 2x - 12 = 0 Let x = 2

2y - 2 (2) - 12 = 0 Remove the brackets

2y - 4 - 12 = 0 Combine

2y - 16 = 0 Add 16 to both sides.

2y = 16 Divide by 2

y = 8

Answer 82

I'll leave the check to you, but the answer of 82 is correct.