document.write( "Question 269458: please help me :using the method of systems of equations i need to solve the folliwing word problem. The sum of the digits of a two-digit number is 12. If the 15 is added to the number, the result is 6 times the units digit. Find the number \n" ); document.write( "
Algebra.Com's Answer #197495 by oberobic(2304)\"\" \"About 
You can put this solution on YOUR website!
In doing sums of digits problems, you have to keep track of the values as well as the counts.
\n" ); document.write( "To illustrate, the number 23 actually is 2 tens plus 3 ones.
\n" ); document.write( "The more general representation of the number 'xy' has the value 10x + y.
\n" ); document.write( "Similarly, the number depicted as 'abc' would have value 100a + 10b + c.
\n" ); document.write( ".
\n" ); document.write( "It is critical that you do not get confused and think 'xy' means x time y!
\n" ); document.write( "It is easy to fall into this trap, especially with a long problem. So beware!
\n" ); document.write( ".
\n" ); document.write( "We are told there is a two-digit number.
\n" ); document.write( "We can call it 'xy'.
\n" ); document.write( "Remember, 'x' is just standing next to 'y'. They are not being multiplied.
\n" ); document.write( ".
\n" ); document.write( "We are told x+y = 12.
\n" ); document.write( "That means x = 12-y and y = 12-x.
\n" ); document.write( ".
\n" ); document.write( "We are told that if you add 15 to the value of the number (i.e., 10x + y), the result is 6y.
\n" ); document.write( "10x + y + 15 = 6y
\n" ); document.write( ".
\n" ); document.write( "Substituting:
\n" ); document.write( "10(12-y) +y + 15= 6y
\n" ); document.write( "120 -10y + y + 15 = 6y
\n" ); document.write( "135 + 9y = 6y
\n" ); document.write( "135 = 15y
\n" ); document.write( "15y = 135
\n" ); document.write( "y = 9
\n" ); document.write( ".
\n" ); document.write( "x = 12-y = 12-9 = 3
\n" ); document.write( ".
\n" ); document.write( "xy = 39
\n" ); document.write( ".
\n" ); document.write( "Check by doing substituting back into the word problem.
\n" ); document.write( "39 can be viewed as xy, which means 3+9 = 12. OK
\n" ); document.write( "39 + 15 = 54, which does = 6*9 = 6y. OK
\n" ); document.write( ".
\n" ); document.write( "So, the number is: 39.
\n" ); document.write( ".
\n" ); document.write( "BUT you teacher may want you to solve the problem using simultaneous equations.
\n" ); document.write( "In that case, we have two equations and two unknowns, so we can do it.
\n" ); document.write( ".
\n" ); document.write( "Eq. 1: x + y = 12
\n" ); document.write( ".
\n" ); document.write( "Eq. 2: 10x + y + 15 = 6y
\n" ); document.write( "Subtracting 6y from both sides
\n" ); document.write( "10x -5y + 15 = 0
\n" ); document.write( "Subtracting 15 from both sides
\n" ); document.write( "10x - 5y = -15
\n" ); document.write( ".
\n" ); document.write( "That gives us:
\n" ); document.write( "x + y = 12
\n" ); document.write( "10x - 5y = -15
\n" ); document.write( "Multiply the first equation by 10
\n" ); document.write( "10x + 10y = 120
\n" ); document.write( "10x - 5y = -15
\n" ); document.write( "Subtracting the second equation from the first
\n" ); document.write( "15y = 135
\n" ); document.write( "Dividing by 15
\n" ); document.write( "y = 9
\n" ); document.write( ".
\n" ); document.write( "And therefore x=3.
\n" ); document.write( ".
\n" ); document.write( "This means the number is 39.
\n" ); document.write( "We checked it above, so there's no need to check it again.
\n" ); document.write( "Done. \n" ); document.write( "
\n" );