document.write( "Question 223613: Solving Equation with Algebraic Fractions:\r
\n" ); document.write( "\n" ); document.write( "7x-6/1-3x=x+2/x-7\r
\n" ); document.write( "\n" ); document.write( "In words: 7x-6 over 1-3x equals x+2 over x-7\r
\n" ); document.write( "\n" ); document.write( "Should have 2 solutions.
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #167135 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Any equation of the form \"a%2Fb+=+x%2Fy\" (in words a fraction equals a fraction) is called a proportion. And proportions can be simplified by cross multiplication. For example, \"a%2Fb+=+x%2Fy\" can be simplified by cross multiplication to \"a%2Ay+=+b%2Ax\".

\n" ); document.write( "Your equation, \"%287x-6%29%2F%281-3x%29+=+%28x%2B2%29%2F%28x-7%29\" is a proportion. So we can cross multiply to simplify it. (Important: Always make sure you have a proportion (two fractions with an equals sign between them) before you use cross multiplication! I've seen literally thousands of errors made by cross multiplying when there is no proportion.)
\n" ); document.write( "So we'll cross multiply:
\n" ); document.write( "\"%287x-6%29%28x-7%29+=+%281-3x%29%28x%2B2%29\"
\n" ); document.write( "and now simplify using FOIL:
\n" ); document.write( "\"7x%5E2+-49x+-6x+%2B+42+=+x+%2B+2+-3x%5E2+-6x\"
\n" ); document.write( "and then add like terms:
\n" ); document.write( "\"7x%5E2+-55x+%2B+42+=+-5x+%2B2+-3x%5E2\"
\n" ); document.write( "Now we'll try to solve this. This is a quadratic equation (because of the \"x%5E2\" terms) so we will make one side of the equation zero. Adding the opposite of the entire right side to both sides gives us:
\n" ); document.write( "\"10x%5E2+-50x+%2B40+=+0\"
\n" ); document.write( "Now we can solve this by factoring or by using the quadratic formula. This factors pretty easily. Always start factoring with the Greatest Common Factor (GCF) if it is not 1. The GCF here is 10 so we'll factor it out:
\n" ); document.write( "\"10%28x%5E2-5x%2B4%29=+0\"
\n" ); document.write( "The trinomial factor also factors:
\n" ); document.write( "\"10%28x-4%29%28x-1%29+=+0\"
\n" ); document.write( "The Zero Product Property says that this or any product that is equal to zero must have a factor that is zero. We have three factors: 10, (x-4) and (x-1). 10 cannot be a zero but the other two could with the \"right\" values for x. So we just solve
\n" ); document.write( "x-4 = 0 or x-1 = 0
\n" ); document.write( "giving:
\n" ); document.write( "x = 4 or x = 1

\n" ); document.write( "Whenever you multiply both sides of an equation by a variable expression, like we did when we cross multiplied, you should either
  • Check your solutions to see that they actually work; or
  • Eliminate any solutions that are not in the domain of the original equation.

\n" ); document.write( "Checking x=4:
\n" ); document.write( "\"%287x-6%29%2F%281-3x%29+=+%28x%2B2%29%2F%28x-7%29\"
\n" ); document.write( "\"%287%284%29-6%29%2F%281-3%284%29%29+=+%28%284%29%2B2%29%2F%28%284%29-7%29\"
\n" ); document.write( "(Note the parentheses used. Whenever you substitute for a variable it is an extremely good habit to place the new value in parentheses!!)
\n" ); document.write( "\"%2828-6%29%2F%281-12%29+=+%28%284%29%2B2%29%2F%28%284%29-7%29\"
\n" ); document.write( "\"%2822%29%2F%28-11%29+=+%286%29%2F%28-3%29\"
\n" ); document.write( "\"-2+=+-2\" Check!

\n" ); document.write( "Checking x=1:
\n" ); document.write( "\"%287x-6%29%2F%281-3x%29+=+%28x%2B2%29%2F%28x-7%29\"
\n" ); document.write( "\"%287%281%29-6%29%2F%281-3%281%29%29+=+%28%281%29%2B2%29%2F%28%281%29-7%29\"
\n" ); document.write( "\"%287-6%29%2F%281-3%29+=+%28%281%29%2B2%29%2F%28%281%29-7%29\"
\n" ); document.write( "\"%281%29%2F%28-2%29+=+%283%29%2F%28-6%29\"
\n" ); document.write( "\"%28-1%29%2F2+=+%28-1%29%2F2\" Check!

\n" ); document.write( "Both solutions, x=4 and x=1, check so we do have two solutions. \n" ); document.write( "
\n" );