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You can put this solution on YOUR website!
let x = the tens digit.
\n" ); document.write( "let y = the ones digit.
\n" ); document.write( "number is 10*x + y
\n" ); document.write( "-----
\n" ); document.write( "product of the two digit number and its tens digit is 54.
\n" ); document.write( "this implies that
\n" ); document.write( "x * (10*x + y) = 54 (first equation)
\n" ); document.write( "-----
\n" ); document.write( "sum of the digits when added to the number gives a result of 36.
\n" ); document.write( "this implies that
\n" ); document.write( "(10*x + y) + x + y = 36 (second equation)
\n" ); document.write( "-----
\n" ); document.write( "it looks like we can eliminate one of the unknowns by solving for y in the second equation.
\n" ); document.write( "second equation again is:
\n" ); document.write( "(10*x + y) + x + y = 36
\n" ); document.write( "remove the parentheses.
\n" ); document.write( "10*x + y + x + y = 36
\n" ); document.write( "combine like terms.
\n" ); document.write( "11*x + 2*y = 36
\n" ); document.write( "subtract 11*x from both sides.
\n" ); document.write( "2*y = 36 - 11*x
\n" ); document.write( "divide both sides by two.
\n" ); document.write( "y = (36 - 11*x)/2
\n" ); document.write( "-----
\n" ); document.write( "first equation again is:
\n" ); document.write( "x * (10*x + y) = 54
\n" ); document.write( "substitute for y.
\n" ); document.write( "x * (10*x + (36-11*x)/2) = 54
\n" ); document.write( "remove the parentheses by multiplying out.
\n" ); document.write( "10*x^2 + (36*x - 11*x^2)/2 = 54
\n" ); document.write( "multiply both sides of equation by 2.
\n" ); document.write( "20*x^2 + 36*x - 11*x^2 = 108
\n" ); document.write( "combine like terms.
\n" ); document.write( "9*x^2 + 36*x = 108
\n" ); document.write( "subtract 108 from both sides.
\n" ); document.write( "9*x^2 + 36*x - 108 = 0
\n" ); document.write( "divide both sides by 9.
\n" ); document.write( "x^2 + 4*x - 12 = 0
\n" ); document.write( "factor the quadratic equation on the left.
\n" ); document.write( "(x-2)*(x+6) = 0
\n" ); document.write( "roots are either:
\n" ); document.write( "x = 2
\n" ); document.write( "or
\n" ); document.write( "x = -6
\n" ); document.write( "-----
\n" ); document.write( "x = -6 doesn't look good since the tens digit needs to be positive.
\n" ); document.write( "solving for y anyway to see what happens.
\n" ); document.write( "second equation reproduced here is:
\n" ); document.write( "(10*x + y) + x + y = 36
\n" ); document.write( "substitute -6 for x.
\n" ); document.write( "10*(-6) + y + (-6) + y = 36
\n" ); document.write( "-60 + y - 6 + y = 36
\n" ); document.write( "combine like terms.
\n" ); document.write( "-66 + 2*y = 36
\n" ); document.write( "add 66 to both sides
\n" ); document.write( "2*y = 102
\n" ); document.write( "y = 51
\n" ); document.write( "impossible since digits must be 0 to 9 only.
\n" ); document.write( "-----
\n" ); document.write( "x = 2 might be the answer.
\n" ); document.write( "second equation reproduced here again is:
\n" ); document.write( "(10*x + y) + x + y = 36
\n" ); document.write( "substituting 2 for x.
\n" ); document.write( "10*2 + y + 2 + y = 36
\n" ); document.write( "multiplying and combining like terms gets
\n" ); document.write( "22 + 2*y = 36
\n" ); document.write( "subtract 22 from both sides.
\n" ); document.write( "2*y = 14
\n" ); document.write( "divided both sides by 2.
\n" ); document.write( "y = 7
\n" ); document.write( "this number is possible.
\n" ); document.write( "-----
\n" ); document.write( "tens digit = 2
\n" ); document.write( "ones digit = 7
\n" ); document.write( "number is 27
\n" ); document.write( "27 * tens digit is 27*2 = 54 which satisfies the first equation requirements.
\n" ); document.write( "27 + 2 + 7 = 36 which satisfies the second equation requirements.
\n" ); document.write( "-----
\n" ); document.write( "number is 27.\r
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