2.3.1: Systems of Linear Equations – Special Cases (Exercises)

Last updated
Save as PDF

Page ID: 37855

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

SECTION 2.3 PROBLEM SET: SYSTEMS OF LINEAR EQUATIONS - SPECIAL CASES

Solve the following inconsistent or dependent systems by using the Gauss-Jordan method.

\begin{aligned} 2 x+6 y&=8 \\ x+3 y&=4 \end{aligned}	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.
\begin{aligned} 2 x-y &=10 \\ -4 x+2 y &=15 \end{aligned}	\begin{aligned} x+y+z&=6 \\ 3 x+2 y+z&=14 \\ 4 x+3 y+2 z&=20 \end{aligned}
\begin{aligned} x+2 y-4 z&=1 \\ 2 x-3 y+8 z&=9 \end{aligned}	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.

SECTION 2.3 PROBLEM SET: SYSTEMS OF LINEAR EQUATIONS - SPECIAL CASES

Solve the following inconsistent or dependent systems by using the Gauss-Jordan method.

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?

\begin{aligned}
x+y+2 z=0 \\
x+2 y+z=0 \\
2 x+3 y+3 z=0
\end{aligned}

Find three solutions to the following system of equations.

\begin{aligned}
x+2 y+z &=12 \\
y &=3
\end{aligned}

\begin{aligned}
x+2 y&=5 \\
2 x+4 y&=k
\end{aligned}

For what values of k does this system of equations have

No solution?
Infinitely many solutions?

$x + 3y - z = 5$

Why is it not possible for a linear system to have exactly two solutions? Explain geometrically.