document.write( "Question 186465: Two perpendicular lines intersect at point (9,2). If the x-intercept of one line is double the x-intercept of the other, find the possible sum(s) of these x- intercepts. \n" ); document.write( "

Algebra.Com's Answer #139827 by josmiceli(19441) $\"\"$ $\"About$

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One line would be
\n" ); document.write( " $\"y+=+mx+%2B+b%5B1%5D\"$
\n" ); document.write( "The other line would be
\n" ); document.write( " $\"y+=+-%281%2Fm%29x+%2B+b%5B2%5D\"$
\n" ); document.write( " $\"b%5B1%5D\"$ and $\"b%5B2%5D\"$ are the y-intercepts
\n" ); document.write( "------------------------
\n" ); document.write( "Let the x-intercepts be $\"x%5B1%5D\"$ and $\"x%5B2%5D\"$
\n" ); document.write( " $\"y+=+0\"$ at the x-intercepts, so
\n" ); document.write( " $\"0+=+m%2Ax%5B1%5D+%2B+b%5B1%5D\"$
\n" ); document.write( " $\"x%5B1%5D+=+-%28b%5B1%5D%2Fm%29\"$
\n" ); document.write( "and
\n" ); document.write( " $\"0+=+-%281%2Fm%29%2Ax%5B2%5D+%2B+b%5B2%5D\"$
\n" ); document.write( " $\"x%5B2%5D+=+m%2Ab%5B2%5D\"$
\n" ); document.write( " $\"x%5B1%5D+%2B+x%5B2%5D+=+m%2Ab%5B2%5D+-+b%5B1%5D%2Fm\"$
\n" ); document.write( "and
\n" ); document.write( " $\"y+=+mx+%2B+b%5B1%5D\"$
\n" ); document.write( " $\"2+=+9m+%2B+b%5B1%5D\"$
\n" ); document.write( " $\"b%5B1%5D+=+2+-+9m\"$
\n" ); document.write( "and
\n" ); document.write( " $\"2+=+-%281%2Fm%29%2A9+%2B+b%5B2%5D\"$
\n" ); document.write( " $\"b%5B2%5D+=+2+%2B+9%2Fm\"$
\n" ); document.write( "Substituting:
\n" ); document.write( " $\"x%5B1%5D+%2B+x%5B2%5D+=+m%2A%282+%2B+9%2Fm%29+-+%282+-+9m%29%2Fm\"$
\n" ); document.write( " $\"x%5B1%5D+%2B+x%5B2%5D+=+2m+%2B+9+-+2%2Fm+%2B+9\"$
\n" ); document.write( " $\"x%5B1%5D+%2B+x%5B2%5D+=+2%28m+-+1%2Fm%29+%2B+18\"$ this is the answer
\n" ); document.write( "Suppose the slopes of the lines are
\n" ); document.write( " $\"m+=+-3\"$
\n" ); document.write( " $\"-%281%2Fm%29+=+1%2F3\"$
\n" ); document.write( "and the solution is (9,2)
\n" ); document.write( "the equations are:
\n" ); document.write( " $\"y+=+-3x+%2B+b%5B1%5D\"$
\n" ); document.write( " $\"2+=+-3%2A9+%2B+b%5B1%5D\"$
\n" ); document.write( " $\"b%5B1%5D+=+2+%2B+27\"$
\n" ); document.write( " $\"b%5B1%5D+=+29\"$
\n" ); document.write( " $\"y+=+-3x+%2B+29\"$ 1st equation
\n" ); document.write( "and
\n" ); document.write( " $\"y+=+%281%2F3%29%2Ax+%2B+b%5B2%5D\"$
\n" ); document.write( " $\"2+=+%281%2F3%29%2A9+%2B+b%5B2%5D\"$
\n" ); document.write( " $\"b%5B2%5D+=+2+-+3\"$
\n" ); document.write( " $\"b%5B2%5D+=+-1\"$
\n" ); document.write( " $\"y+=+%281%2F3%29%2Ax+-+1\"$ 2nd equation
\n" ); document.write( "Setting $\"y+=+0\"$ in both will find the x-intercepts
\n" ); document.write( " $\"y+=+-3x+%2B+29\"$
\n" ); document.write( " $\"0+=+-3%2Ax%5B1%5D+%2B+29\"$
\n" ); document.write( " $\"x%5B1%5D+=+29%2F3\"$
\n" ); document.write( "and
\n" ); document.write( " $\"0+=+%281%2F3%29%2Ax%5B2%5D+-+1\"$
\n" ); document.write( " $\"x%5B2%5D+=+3\"$
\n" ); document.write( "and
\n" ); document.write( " $\"x%5B1%5D+%2B+x%5B2%5D+=+2%28m+-+1%2Fm%29+%2B+18\"$
\n" ); document.write( " $\"29%2F3+%2B+3+=+2%28-3+-+1%2F%28-3%29%29+%2B+18\"$
\n" ); document.write( " $\"29%2F3+%2B+9%2F3+=+2%28-3+%2B+1%2F3%29+%2B+18\"$
\n" ); document.write( " $\"29%2F3+%2B+9%2F3+=+-6+%2B+2%2F3+%2B+18\"$
\n" ); document.write( "Multiply both sides by $\"3\"$
\n" ); document.write( " $\"29+%2B+9+=+-18+%2B+2+%2B+54\"$
\n" ); document.write( " $\"38+=+-16+%2B+54\"$
\n" ); document.write( " $\"38+=+38\"$
\n" ); document.write( "OK
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