+564 The sum of the digits of a two-digit counting number is 10. If the digits are reversed, the new number is one less than twice the original number. What was the original number?
A + B=10
2[10A + B] - 1=10B + A
So the original number is=37
Solve the following system:
{A+B = 10 | (equation 1)
2 (10 A+B)-1 = A+10 B | (equation 2)
Express the system in standard form:
{A+B = 10 | (equation 1)
19 A-8 B = 1 | (equation 2)
Swap equation 1 with equation 2:
{19 A-8 B = 1 | (equation 1)
A+B = 10 | (equation 2)
Subtract 1/19 × (equation 1) from equation 2:
{19 A-8 B = 1 | (equation 1)
0 A+(27 B)/19 = 189/19 | (equation 2)
Multiply equation 2 by 19/27:
{19 A-8 B = 1 | (equation 1)
0 A+B = 7 | (equation 2)
Add 8 × (equation 2) to equation 1:
{19 A+0 B = 57 | (equation 1)
0 A+B = 7 | (equation 2)
Divide equation 1 by 19:
{A+0 B = 3 | (equation 1)
0 A+B = 7 | (equation 2)
Collect results:
Answer: | A = 3 and B = 7
