# 9.6.3. Solving Two Equations in Two Variables, One Is of the First Degree, and the Other Is of the Second Degree

In a two-digit number, the units digit is times times the tens digit. Determine the original number, given that the product of the two digits is of its value.

• A28
• B82
• C92
• D83

### Example

In a two-digit number, the units digit is times times the tens digit. Determine the original number, given that the product of the two digits is of its value.

### Solution

Denote the tens digit of the number by and the units digit by . If the units digit is 4 times the tens digit, then

If the product of the digits is of the number's value, then

This pair of equations can be solved by substitution. Substituting the expression for into the second order equation gives

This equation can be solved by factorisation.

But as is a two-digit number, its first digit cannot be 0. Therefore, .

Now substitute this value of into the first order equation to find .

The two-digit number is therefore 28.

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