document.write( "Question 782850: $\"+9n%2B4=5n%2B18+\"$
\n" ); document.write( " $\"+%28-4%299n%2B4=5n%2B18%28-4%29+\"$ \n" ); document.write( "

Algebra.Com's Answer #476585 by KMST(5328) $\"\"$ $\"About$

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$\"9n%2B4=5n%2B18\"$ <--> $\"9n%2B4-4=5n%2B18-4\"$ <--> $\"9n=5n%2B14\"$ <--> $\"9n-5n=5n%2B14-5n\"$ <--> $\"4n=14\"$ <--> $\"4n%2F4=14%2F4\"$ <--> $\"n=14%2F4\"$ <--> $\"highlight%28n=7%2F2%29\"$
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\n" ); document.write( "To solve an equation like $\"9n%2B4=5n%2B18\"$ , we tranform it into simpler and simpler equivalent equations in several steps.
\n" ); document.write( "(Equations are called equivalent when they have exactly the same solutions).
\n" ); document.write( "An equation can be transformed into an equivalent equation by:
\n" ); document.write( "adding the same number (or expression) to both sides of the equal sign, or
\n" ); document.write( "subtracting the same number (or expression) from both sides of the equal sign, or
\n" ); document.write( "multiplying times the same non-zero number (or expression) both sides of the equal sign, or
\n" ); document.write( "dividinging by the same non-zero number (or expression) both sides of the equal sign.
\n" ); document.write( "The steps used above were:
\n" ); document.write( "subtract $\"4\"$ from both sides,
\n" ); document.write( "subtract $\"5n\"$ from both sides, and
\n" ); document.write( "divide both sides by $\"4\"$ .
\n" ); document.write( "Why those steps in that order?
\n" ); document.write( "Because we have to \"undo\" all the bad tricks played on $\"n\"$ until we get to an equation looking like $\"n=%22...%22\"$ .
\n" ); document.write( "Such an equation is the solution.
\n" ); document.write( "The rules about order of operations were established long ago so that we could all agree on what we meant when we talk math in \"full sentences\".
\n" ); document.write( "According to the rules of order of operations, in the expression $\"9n%2B4\"$ , $\"n\"$ was \"dressed up\" by first multiplying times $\"9\"$ , and then adding $\"4\"$ .
\n" ); document.write( "We need to \"undress\" it.
\n" ); document.write( "To do that, we start by undoing the last \"dressing\" step first.
\n" ); document.write( "So we undo the adding $\"4\"$ by subtracting $\"4\"$ .
\n" ); document.write( "After that the left side is simpler, just $\"9n\"$ , but we have $\"n\"$ on both sides.
\n" ); document.write( "By subtracting $\"5n\"$ from both sides, we can get a simpler right side (just $\"14\"$ ), and end up with $\"n\"$ appearing only on the left side.
\n" ); document.write( "That leaves us with $\"4n=14\"$ , where $\"n\"$ is multiplied times $\"4\"$ .
\n" ); document.write( "We \"undo that multiplication by dividing by $\"4\"$ .
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\n" ); document.write( "NOTE 1:
\n" ); document.write( " $\"+9n%2B4=5n%2B18+\"$ <--> $\"+%28-4%29%289n%2B4%29=%285n%2B18%29%28-4%29+\"$
\n" ); document.write( "shows two equivalent equations, but does not help in solving the first equation.
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\n" ); document.write( "NOTE 2:
\n" ); document.write( "The equation $\"%28-4%299n%2B4=5n%2B18%28-4%29\"$ is not equivalent, because the $\"%28-4%29\"$ is not multiplied times the entire left side, and times entire right side of the equation.
\n" ); document.write( "It is multiplying only $\"9n\"$ on the left, and only $\"18\"$ on the right. \n" ); document.write( "

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