1.1.1: Exercises 1.1

Last updated
Save as PDF

Page ID: 63608

$ \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}}$

In Exercises $\PageIndex{1}$ - $\PageIndex{10}$, state whether or not the given equation is linear.

Exercise $\PageIndex{1}$

$x+y+z = 10$

Answer: Linear

Exercise $\PageIndex{2}$

$xy + yz+ xz = 1$

Answer: Nonlinear

Exercise $\PageIndex{3}$

$-3x + 9 = 3y - 5z+ x-7$

Answer: Linear

Exercise $\PageIndex{4}$

$\sqrt{5}y + \pi x =-1$

Answer: Linear

Exercise $\PageIndex{5}$

$(x-1)(x+1) = 0$

Answer: Nonlinear

Exercise $\PageIndex{6}$

$\sqrt{x_1^2+x_2^2} = 25$

Answer: Nonlinear

Exercise $\PageIndex{7}$

$x_1 + y + t = 1$

Answer: Linear

Exercise $\PageIndex{8}$

$\frac{1}{x} + 9 = 3\cos(y) - 5z$

Answer: Nonlinear

Exercise $\PageIndex{9}$

$\cos(15)y + \frac{x}{4} =-1$

Answer: Linear

Exercise $\PageIndex{10}$

$2^x + 2^y = 16$

Answer: Nonlinear

In Exercises $\PageIndex{11}$ - $\PageIndex{14}$, solve the system of linear equations.

Exercise $\PageIndex{11}$

$\begin{array}{ccccc} x&+&y&=&-1\\ 2x&-&3y&=&8\\ \end{array}$

Answer: $x = 1, y=-2$

Exercise $\PageIndex{12}$

$\begin{array}{ccccc} 2x&-&3y&=&3\\ 3x&+&6y&=&8\\ \end{array}$

Answer: $x = 2, y=\frac13$

Exercise $\PageIndex{13}$

$\begin{array}{ccccccc} x&-&y&+&z&=&1\\ 2x&+&6y&-&z&=&-4\\ 4x&-&5y&+&2z&=&0\\ \end{array}$

Answer: $x = -1, y=0,z=2$

Exercise $\PageIndex{14}$

$\begin{array}{ccccccc} x&+&y&-&z&=&1\\ 2x&+&y&&&=&2\\ &&y&+&2z&=&0\\ \end{array}$

Answer: $x =1,\ y=0,\ z=0$

Exercise $\PageIndex{15}$

A farmer looks out his window at his chickens and pigs. He tells his daughter that he sees 62 heads and 190 legs. How many chickens and pigs does the farmer have?

Answer: 29 chickens and 33 pigs

Exercise $\PageIndex{16}$

A lady buys 20 trinkets at a yard sale. The cost of each trinket is either $0.30 or $0.65. If she spends $8.80, how many of each type of trinket does she buy?

Answer: 12 $0.30 trinkets, 8 $0.65 trinkets

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

Exercise \(\PageIndex{7}\)

Exercise \(\PageIndex{8}\)

Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\)

Exercise \(\PageIndex{11}\)

Exercise \(\PageIndex{12}\)

Exercise \(\PageIndex{13}\)

Exercise \(\PageIndex{14}\)

Exercise \(\PageIndex{15}\)

Exercise \(\PageIndex{16}\)