The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?
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Step-by-Step Solution
Step 1: Represent the two-digit number
We can represent a two-digit number using its tens digit, x, and its units digit, y. So, the number can be written as 10x + y (e.g., if x=5 and y=6, the number is 10(5)+6=56). The problem refers to this representation in <ref_1>.
Step 2: Represent the number with reversed digits
When the digits are reversed, the units digit becomes the tens digit and the tens digit becomes the units digit. So, y becomes the tens digit and x becomes the units digit, forming the number 10y + x.
Step 3: Form the first equation based on the sum
The problem states that the sum of the original two-digit number and the number obtained by reversing its digits is 66. We set up an equation reflecting this condition.
Step 4: Simplify the first equation
Combine like terms in the sum equation. We have 10x+x=11x and y+10y=11y. Then, divide the entire equation by 11 to simplify it, resulting in Equation 1 as x+y=6.
Step 5: Form the second equation based on digit difference
The problem states that the digits differ by 2. This means there are two possibilities: either x is greater than y (x−y=2), or y is greater than x (y−x=2). We will solve for both cases as stated in the example in <ref_1>.
Step 6: Solve Case 1: x−y=2
We now have a system of two linear equations: x+y=6 and x−y=2. We will solve this system using the elimination method, as demonstrated in example 9 of <ref_2>.
Step 7: Add the equations for Case 1
Adding Equation 1 and Equation 2a eliminates the y variable, allowing us to solve for x. We find that 2x=8, which simplifies to x=4.
Step 8: Substitute x to find y for Case 1
Substitute the value of x=4 into Equation 1 (x+y=6) to find the value of y. This gives us 4+y=6, which results in y=2.
Step 9: Determine the number for Case 1
With x=4 and y=2, the original two-digit number is 10(4)+2=42. We can verify this: 42+24=66, and 4−2=2. This matches the conditions.
Step 10: Solve Case 2: y−x=2
For the second case, we consider y−x=2, which can be rewritten as −x+y=2. We again use the elimination method with Equation 1.
Step 11: Add the equations for Case 2
Adding Equation 1 and Equation 2b eliminates the x variable, allowing us to solve for y. We find that 2y=8, which simplifies to y=4.
Step 12: Substitute y to find x for Case 2
Substitute the value of y=4 into Equation 1 (x+y=6) to find the value of x. This gives us x+4=6, which results in x=2.
Step 13: Determine the number for Case 2
With x=2 and y=4, the original two-digit number is 10(2)+4=24. We can verify this: 24+42=66, and 4−2=2. This also matches the conditions.
Step 14: Count the number of solutions
We found two possible numbers that satisfy both given conditions. These numbers are 42 and 24.