SECTION 2.3 PROBLEM SET: SYSTEMS OF LINEAR EQUATIONS - SPECIAL CASES
Solve the following inconsistent or dependent systems by using the Gauss-Jordan method.
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\begin{aligned}
2 x+6 y&=8 \\
x+3 y&=4
\end{aligned}
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The sum of digits of a two digit number is 9. The sum of the number and the number obtained by interchanging the digits is 99. Find the number.
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\begin{aligned}
2 x-y &=10 \\
-4 x+2 y &=15
\end{aligned}
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\begin{aligned}
x+y+z&=6 \\
3 x+2 y+z&=14 \\
4 x+3 y+2 z&=20
\end{aligned}
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\begin{aligned}
x+2 y-4 z&=1 \\
2 x-3 y+8 z&=9
\end{aligned}
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Jessica has a collection of 15 coins consisting of nickels, dimes and quarters. If the total worth of the coins is $1.80, how many are there of each? Find all three solutions.
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SECTION 2.3 PROBLEM SET: SYSTEMS OF LINEAR EQUATIONS - SPECIAL CASES
Solve the following inconsistent or dependent systems by using the Gauss-Jordan method.
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A company is analyzing sales reports for three products: products X, Y, Z. One report shows that a combined total of 20,000 of items X, Y, and Z were sold. Another report shows that the sum of the number of item Z sold and twice the number of item X sold equals 10,000. Also item X has 5,000 more items sold than item Y. Are these reports consistent?
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\begin{aligned}
x+y+2 z=0 \\
x+2 y+z=0 \\
2 x+3 y+3 z=0
\end{aligned}
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Find three solutions to the following system of equations.
\begin{aligned}
x+2 y+z &=12 \\
y &=3
\end{aligned}
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\begin{aligned}
x+2 y&=5 \\
2 x+4 y&=k
\end{aligned}
For what values of k does this system of equations have
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No solution?
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Infinitely many solutions?
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\(x + 3y - z = 5\)
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Why is it not possible for a linear system to have exactly two solutions? Explain geometrically.
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