document.write( "Question 109400This question is from textbook
\n" ); document.write( ": The sum of two numbers is 111, and their difference is 63.
\n" ); document.write( "What are the two numbers?\r
\n" ); document.write( "\n" ); document.write( "*what is the formula? \n" ); document.write( "
Algebra.Com's Answer #79755 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
The sum of two numbers can be written as \"x%2By=111\" and the difference can be written as \"x-y=63\" which makes the system\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\"x%2By=111\"
\n" ); document.write( "\"x-y=63\"\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "\n" ); document.write( "
Solved by pluggable solver: Solving a linear system of equations by subsitution

\n" ); document.write( "
\n" ); document.write( " Lets start with the given system of linear equations
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax%2B1%2Ay=111\"
\n" ); document.write( " \"1%2Ax-1%2Ay=63\"
\n" ); document.write( "
\n" ); document.write( " Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.
\n" ); document.write( "
\n" ); document.write( " Solve for y for the first equation
\n" ); document.write( "
\n" ); document.write( " \"1%2Ay=111-1%2Ax\"Subtract \"1%2Ax\" from both sides
\n" ); document.write( "
\n" ); document.write( " \"y=%28111-1%2Ax%29\" Divide both sides by 1.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Which breaks down and reduces to
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y=111-1%2Ax\" Now we've fully isolated y
\n" ); document.write( "
\n" ); document.write( " Since y equals \"111-1%2Ax\" we can substitute the expression \"111-1%2Ax\" into y of the 2nd equation. This will eliminate y so we can solve for x.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax%2B-1%2Ahighlight%28%28111-1%2Ax%29%29=63\" Replace y with \"111-1%2Ax\". Since this eliminates y, we can now solve for x.
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax-1%2A%28111%29-1%28-1%29x=63\" Distribute -1 to \"111-1%2Ax\"
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax-111%2B1%2Ax=63\" Multiply
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax-111%2B1%2Ax=63\" Reduce any fractions
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax%2B1%2Ax=63%2B111\"Add \"111\" to both sides
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax%2B1%2Ax=174\" Combine the terms on the right side
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"2%2Ax=174\" Now combine the terms on the left side.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"cross%28%281%2F2%29%282%2F1%29%29x=%28174%2F1%29%281%2F2%29\" Multiply both sides by \"1%2F2\". This will cancel out \"2%2F1\" and isolate x
\n" ); document.write( "
\n" ); document.write( " So when we multiply \"174%2F1\" and \"1%2F2\" (and simplify) we get
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"x=87\" <---------------------------------One answer
\n" ); document.write( "
\n" ); document.write( " Now that we know that \"x=87\", lets substitute that in for x to solve for y
\n" ); document.write( "
\n" ); document.write( " \"1%2887%29-1%2Ay=63\" Plug in \"x=87\" into the 2nd equation
\n" ); document.write( "
\n" ); document.write( " \"87-1%2Ay=63\" Multiply
\n" ); document.write( "
\n" ); document.write( " \"-1%2Ay=63-87\"Subtract \"87\" from both sides
\n" ); document.write( "
\n" ); document.write( " \"-1%2Ay=-24\" Combine the terms on the right side
\n" ); document.write( "
\n" ); document.write( " \"cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-24%2F1%29%281%2F-1%29\" Multiply both sides by \"1%2F-1\". This will cancel out -1 on the left side.
\n" ); document.write( "
\n" ); document.write( " \"y=-24%2F-1\" Multiply the terms on the right side
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y=24\" Reduce
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " So this is the other answer
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y=24\"<---------------------------------Other answer
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " So our solution is
\n" ); document.write( "
\n" ); document.write( " \"x=87\" and \"y=24\"
\n" ); document.write( "
\n" ); document.write( " which can also look like
\n" ); document.write( "
\n" ); document.write( " (\"87\",\"24\")
\n" ); document.write( "
\n" ); document.write( " Notice if we graph the equations (if you need help with graphing, check out this solver)
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax%2B1%2Ay=111\"
\n" ); document.write( " \"1%2Ax-1%2Ay=63\"
\n" ); document.write( "
\n" ); document.write( " we get
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " graph of \"1%2Ax%2B1%2Ay=111\" (red) and \"1%2Ax-1%2Ay=63\" (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " and we can see that the two equations intersect at (\"87\",\"24\"). This verifies our answer.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " -----------------------------------------------------------------------------------------------
\n" ); document.write( " Check:
\n" ); document.write( "
\n" ); document.write( " Plug in (\"87\",\"24\") into the system of equations
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Let \"x=87\" and \"y=24\". Now plug those values into the equation \"1%2Ax%2B1%2Ay=111\"
\n" ); document.write( "
\n" ); document.write( " \"1%2A%2887%29%2B1%2A%2824%29=111\" Plug in \"x=87\" and \"y=24\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"87%2B24=111\" Multiply
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"111=111\" Add
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"111=111\" Reduce. Since this equation is true the solution works.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " So the solution (\"87\",\"24\") satisfies \"1%2Ax%2B1%2Ay=111\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Let \"x=87\" and \"y=24\". Now plug those values into the equation \"1%2Ax-1%2Ay=63\"
\n" ); document.write( "
\n" ); document.write( " \"1%2A%2887%29-1%2A%2824%29=63\" Plug in \"x=87\" and \"y=24\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"87-24=63\" Multiply
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"63=63\" Add
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"63=63\" Reduce. Since this equation is true the solution works.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " So the solution (\"87\",\"24\") satisfies \"1%2Ax-1%2Ay=63\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Since the solution (\"87\",\"24\") satisfies the system of equations
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"1%2Ax%2B1%2Ay=111\"
\n" ); document.write( " \"1%2Ax-1%2Ay=63\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " this verifies our answer.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "

\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So our two numbers are 87 and 24 \n" ); document.write( "
\n" );