Let's solve the given problem step by step.
### Part (a): Express the Given Conditions in the Form of Linear Equations
1. Let [tex]\( x \)[/tex] be the digit in the tens place of the number.
2. Let [tex]\( y \)[/tex] be the digit in the units place of the number.
#### Condition 1:
The digit in the units' place ([tex]\( y \)[/tex]) is one-fourth of the digit in the tens place ([tex]\( x \)[/tex]):
[tex]\[ y = \frac{x}{4} \][/tex]
#### Condition 2:
When the digits are reversed, the new number formed is 54 less than the original number. If the original number is [tex]\( 10x + y \)[/tex], then the reversed number will be [tex]\( 10y + x \)[/tex]. Therefore:
[tex]\[ 10y + x = 10x + y - 54 \][/tex]
So, the equations we have are:
1. [tex]\( y = \frac{x}{4} \)[/tex]
2. [tex]\( 10y + x = 10x + y - 54 \)[/tex]
### Part (b): Find the Number
Now, let's solve these equations to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
#### Using the first equation [tex]\( y = \frac{x}{4} \)[/tex]:
This implies [tex]\( y = \frac{x}{4} \)[/tex].
#### Substitute [tex]\( y = \frac{x}{4} \)[/tex] in the second equation:
[tex]\[ 10\left(\frac{x}{4}\right) + x = 10x + \left(\frac{x}{4}\right) - 54 \][/tex]
Simplify:
[tex]\[ \frac{10x}{4} + x = 10x + \frac{x}{4} - 54 \][/tex]
[tex]\[ \frac{10x + 4x}{4} = 10x + \frac{x}{4} - 54 \][/tex]
[tex]\[ \frac{14x}{4} = 10x + \frac{x}{4} - 54 \][/tex]
Multiply everything by [tex]\( 4 \)[/tex] to clear the denominators:
[tex]\[ 14x = 40x + x - 216 \][/tex]
[tex]\[ 14x = 41x - 216 \][/tex]
Rearrange to isolate [tex]\( x \)[/tex]:
[tex]\[ 14x - 41x = -216 \][/tex]
[tex]\[ -27x = -216 \][/tex]
[tex]\[ x = 8 \][/tex]
Now, substitute [tex]\( x = 8 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = \frac{8}{4} \][/tex]
[tex]\[ y = 2 \][/tex]
Therefore, the original number is:
[tex]\[ 10x + y = 10(8) + 2 = 80 + 2 = 82 \][/tex]
### Part (c): Find the Reverse Number
The reverse number is formed by reversing the digits:
[tex]\[ 10y + x = 10(2) + 8 = 20 + 8 = 28 \][/tex]
### Final Answer
- The original number is 82.
- The reverse number is 28.
