document.write( "Question 1085335: 4x+y=3 and (2k-1)x +(k-1)y=2k+1 is inconsistent ,find k \n" ); document.write( "

Algebra.Com's Answer #699408 by KMST(5328) $\"\"$ $\"About$

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For the system in the problem to be inconsistent,
\n" ); document.write( "the coefficients of x and y in $\"%282k-1%29x+%2B%28k-1%29y=2k%2B1\"$
\n" ); document.write( "have to equal to the coefficients in $\"4x%2By=3\"$ multiplied by a certain number,
\n" ); document.write( "but the same cannot be true of the terms not connected to variables:
\n" ); document.write( " $\"2k%2B1\"$ and $\"3\"$ .
\n" ); document.write( "So, it must be that $\"%282k-1%29%2F4=%28k-1%29%2F1\"$ .
\n" ); document.write( "The ratio is $\"k-1\"$ ,
\n" ); document.write( "but that ratio must be different from $\"%282k%2B1%29%2F3\"$
\n" ); document.write( "Solving:
\n" ); document.write( " $\"%282k-1%29%2F4=%28k-1%29%2F1\"$
\n" ); document.write( " $\"2k-1=4%28k-1%29\"$
\n" ); document.write( " $\"2k-1=4k-4\"$
\n" ); document.write( " $\"-1%2B4=4k-2k\"$
\n" ); document.write( " $\"3=2k\"$
\n" ); document.write( " $\"3%2F2=k\"$
\n" ); document.write( "With $\"k=3%2F2\"$ , the coefficients' ratio is $\"k-1=3%2F2-1=1%2F2\"$ ,
\n" ); document.write( "and $\"%282k%2B1%29%2F3=%282%28%223+%2F+2%22%29%2B1%29%2F3=%283%2B1%29%2F3=4%2F3\"$ is not the same.
\n" ); document.write( "So, $\"highlight%28k=3%2F2%29\"$ makes the system inconsistent.
\n" ); document.write( "
\n" ); document.write( "LONG-WINDED EXPLANATION:
\n" ); document.write( "Linear equations graph as straight lines.
\n" ); document.write( "If the lines intersect, there is a solution to the system:
\n" ); document.write( "the set of coordinates for the intersection point.
\n" ); document.write( "Lines that intersect do not have the same slope.
\n" ); document.write( "The lines could have the same slope,
\n" ); document.write( "as in $\"system%284x%2By=3%2C4x%2By=5%29\"$ , or $\"system%284x%2By=3%2C4x%2By=3%29\"$ , or $\"system%284x%2By=3%2C8x%2B2y=6%29\"$ , or $\"system%284x%2By=3%2C8x%2B2y=10%29\"$ .
\n" ); document.write( "If the slopes are the same it could be that the lines are really the same line,
\n" ); document.write( "as is the case with $\"4x%2By=3\"$ and $\"8x%2B2y=6\"$ ,
\n" ); document.write( "and in that case, there is an infinite number of solutions:
\n" ); document.write( "the sets of coordinates for each of the infinite number of points on the line.
\n" ); document.write( "The other possibility, if the slopes are the same is cases like $\"system%284x%2By=3%2C4x%2By=5%29\"$ ,
\n" ); document.write( "where it is obvious that the two equations cannot be true at the same time.
\n" ); document.write( "There are no solutions, and the system of linear equations is called inconsistent.
\n" ); document.write( "If $\"4x%2By\"$ is $\"3\"$ it cannot also be $\"5\"$ for the same x and y values;
\n" ); document.write( "one equation is not consistent with the other.
\n" ); document.write( "Those equations represent parallel lines, with no point in common.
\n" ); document.write( "It is the same case with $\"system%284x%2By=3%2C8x%2B2y=10%29\"$ ,
\n" ); document.write( "although it is not so obvious. $\"8x%2B2y=10\"$ is equivalent to $\"4x%2By=5\"$ .
\n" ); document.write( "The coefficients of $\"8x%2B2y=10\"$ are 2 times those of $\"4x%2By=3\"$ ,
\n" ); document.write( "but the number 10 (not attached to any variable, on the other side of the equal sign)
\n" ); document.write( "is not 2 times 3. \n" ); document.write( "

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