document.write( "Question 1206698: Find all values of k for which the given the augmented matrix corresponds to a consistent linear system
\n" ); document.write( " K 1 -2
\n" ); document.write( " 4 -1 2 \n" ); document.write( "

Algebra.Com's Answer #844291 by math_tutor2020(3817) $\"\"$ $\"About$

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\n" ); document.write( "Definition: A consistent system has at least one solution.
\n" ); document.write( "In contrast, an inconsistent system has no solutions.\r
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\n" ); document.write( "\n" ); document.write( "Let's consider a real number k such that $\"k+%3C%3E+0\"$ and $\"k+%3C%3E+-4\"$
\n" ); document.write( "These restrictions on k are to avoid division by zero errors in the matrix row reduction shown below.
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k	1	-2
4	-1	2

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1	1/k	-2/k	(1/k)*R1 --> R1
4	-1	2

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1	1/k	-2/k
0	-(k+4)/k	(2k+8)/k	R2 - 4R1 --> R2

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1	1/k	-2/k
0	1	-2	(-k/(k+4))*R2 --> R2

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1	0	0	R1 - (1/k)*R2 --> R1
0	1	-2

\n" ); document.write( "The matrix is now in reduced row echelon form (RREF)
\n" ); document.write( "The solution is (x,y) = (0,-2) to prove this system is consistent.\r
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\n" ); document.write( "\n" ); document.write( "Now consider k = 0.
\n" ); document.write( "kx+y = -2
\n" ); document.write( "0*x+y = -2
\n" ); document.write( "y = -2
\n" ); document.write( "Then,
\n" ); document.write( "4x-y = 2
\n" ); document.write( "4x-(-2) = 2
\n" ); document.write( "4x+2 = 2
\n" ); document.write( "4x = 2-2
\n" ); document.write( "4x = 0
\n" ); document.write( "x = 0/4
\n" ); document.write( "x = 0
\n" ); document.write( "We arrive at (x,y) = (0,-2) again.
\n" ); document.write( "The system is consistent when k = 0.\r
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\n" ); document.write( "\n" ); document.write( "Now consider k = -4.
\n" ); document.write( "kx+y = -2
\n" ); document.write( "-4x+y = -2
\n" ); document.write( "We go from this system
\n" ); document.write( " $\"system%28kx%2By+=+-2%2C4x-y=2%29\"$
\n" ); document.write( "to this system
\n" ); document.write( " $\"system%28-4x%2By+=+-2%2C4x-y=2%29\"$
\n" ); document.write( "Adding straight down yields 0x+0y = 0 or in short 0 = 0.
\n" ); document.write( "This system is consistent when k = -4.
\n" ); document.write( "Unlike the other cases, we get infinitely many solutions here. Each solution is of the form (x,y) = (x, 4x-2)
\n" ); document.write( "Note x = 0 leads to y = -2 to show that (0,-2) is one of the infinitely many solutions here. \r
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\n" ); document.write( "\n" ); document.write( "Summary:\r
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\n" ); document.write( "\n" ); document.write( "We conclude that the system $\"system%28kx%2By=-2%2C4x-y=2%29\"$ is consistent for any real number k.
\n" ); document.write( "Meaning that this system will have at least one solution.
\n" ); document.write( "If k = -4 then it has infinitely many solutions of the form (x,4x-2). Otherwise it will have exactly one solution which is (0,-2).\r
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\n" ); document.write( "\n" ); document.write( "Here is an interactive Desmos graph.
\n" ); document.write( "https://www.desmos.com/calculator/mh8pmourgs
\n" ); document.write( "Move the slider around for the k value to see the red line rotating around. The center of rotation is (0,-2). When k = -4 the two lines overlap.
\n" ); document.write( "It is impossible to pick a value of k to make the system inconsistent.
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