• Home
  • Textbooks
  • Elementary Linear Algebra: Applications Version
  • Systems of Linear Equations and Matrices

Elementary Linear Algebra: Applications Version

Howard Anton, Chris Rorres

Chapter 1

Systems of Linear Equations and Matrices - all with Video Answers

Educators

+ 5 more educators

Section 1

Introduction to Systems of Linear Equations

01:13

Problem 1

In each part, determine whether the equation is linear in $x_{1}$ $x_{2},$ and $x_{3}$.
(a) $x_{1}+5 x_{2}-\sqrt{2} x_{3}=1$
(b) $x_{1}+3 x_{2}+x_{1} x_{3}=2$
(c) $x_{1}=-7 x_{2}+3 x_{3}$
(d) $x_{1}^{-2}+x_{2}+8 x_{3}=5$
(e) $x_{1}^{3 / 5}-2 x_{2}+x_{3}=4$
(f) $\pi x_{1}-\sqrt{2} x_{2}=7^{1 / 3}$

Breanna Ollech
Breanna Ollech
Numerade Educator
01:59

Problem 2

In each part, determine whether the equation is linear in $x$ and $y$
(a) $2^{1 / 3} x+\sqrt{3} y=1$
(b) $2 x^{1 / 3}+3 \sqrt{y}=1$
(c) $\cos \left(\frac{\pi}{7}\right) x-4 y=\log 3$
(d) $\frac{\pi}{7} \cos x-4 y=0$
(e) $x y=1$
(f) $y+7=x$

WM
William Mead
Numerade Educator
02:18

Problem 3

Using the notation of Formula $(7),$ write down a general linear system of
(a) two equations in two unknowns.
(b) three equations in three unknowns.
(c) two equations in four unknowns.

Abby Kennedy
Abby Kennedy
Numerade Educator
03:00

Problem 4

Write down the augmented matrix for each of the linear systems in Exercise 3.

Nicolas Ventura
Nicolas Ventura
Numerade Educator
04:13

Problem 5

In each part find a linear system in the unknowns $x_{1}, x_{2}, x_{3}, \ldots,$ that corresponds to the given augmented matrix.
$$\text { (a) }\left[\begin{array}{ccc} 2 & 0 & 0 \\ 3 & -4 & 0 \\ 0 & 1 & 1 \end{array}\right]$$
$$\text { (b) }\left[\begin{array}{rrrr} 3 & 0 & -2 & 5 \\ 7 & 1 & 4 & -3 \\ 0 & -2 & 1 & 7 \end{array}\right]$$

Ryo Kudo
Ryo Kudo
Numerade Educator
03:22

Problem 6

In each part find a linear system in the unknowns $x_{1}, x_{2}, x_{3}, \ldots,$ that corresponds to the given augmented matrix.
$$\text { (a) }\left[\begin{array}{ccccc} 0 & 3 & -1 & -1 & -1 \\ 5 & 2 & 0 & -3 & -6 \end{array}\right]$$
$$\text { (b) }\left[\begin{array}{rrrrr} 3 & 0 & 1 & -4 & 3 \\ -4 & 0 & 4 & 1 & -3 \\ -1 & 3 & 0 & -2 & -9 \\
0 & 0 & 0 & -1 & -2 \end{array}\right]$$

Pratul Kasote
Pratul Kasote
Numerade Educator
03:29

Problem 7

Find the augmented matrix for the linear system.
(a) $-2 x_{1}=6$ $3 x_{1}=8$ $9 x_{1}=-3$
(b) $6 x_{1}-x_{2}+3 x_{3}=4$ $5 x_{2}-x_{3}=1$
$\begin{array}{llll}\text { (c) } & 2 x_{2} & -3 x_{4}+x_{5}= & 0\end{array}$ $-3 x_{1}-x_{2}+x_{3}$ $6 x_{1}+2 x_{2}-x_{3}+2 x_{4}-3 x_{5}=6$

Pratul Kasote
Pratul Kasote
Numerade Educator
03:24

Problem 8

Find the augmented matrix for the linear system.
(a) $3 x_{1}-2 x_{2}=-1$ $4 x_{1}+5 x_{2}=3$ $7 x_{1}+3 x_{2}=2$
(b) $2 x_{1} \quad+2 x_{3}=1$ $3 x_{1}-x_{2}+4 x_{3}=7$ $6 x_{1}+x_{2}-x_{3}=0$
(c) $x_{1} \quad=1$ $x_{2} \quad=2$ $x_{3}=3$

Pratul Kasote
Pratul Kasote
Numerade Educator
02:18

Problem 9

In each part, determine whether the given 3 -tuple is a solution of the linear system $$\begin{aligned} 2 x_{1}-4 x_{2}-x_{3} &=1 \\ x_{1}-3 x_{2}+x_{3} &=1 \\ 3 x_{1}-5 x_{2}-3 x_{3} &=1 \end{aligned}$$
(a) (3,1,1)
(b) (3,-1,1)
(c) (13,5,2)
(d) $\left(\frac{13}{2}, \frac{5}{2}, 2\right)$
(e) (17,7,5)

Pratul Kasote
Pratul Kasote
Numerade Educator
06:09

Problem 10

In each part, determine whether the given 3-tuple is a solution of the linear system
$$\begin{array}{r}x+2 y-2 z=3 \\3 x-y+z=1 \\-x+5 y-5 z=5\end{array}$$
(a) $\left(\frac{5}{7}, \frac{8}{7}, 1\right)$
(b) $\left(\frac{5}{7}, \frac{8}{7}, 0\right)$
(c) (5,8,1)
(d) $\left(\frac{5}{7}, \frac{10}{7}, \frac{2}{7}\right)$
(e) $\left(\frac{5}{7}, \frac{22}{7}, 2\right)$

Lauren Shelton
Lauren Shelton
Numerade Educator
03:57

Problem 11

In each part, solve the linear system, if possible, and use the result to determine whether the lines represented by the equations in the system have zero, one, or infinitely many points of intersection. If there is a single point of intersection, give its coordinates, and if there are infinitely many, find parametric equations for them.
(a) $3 x-2 y=4$ $6 x-4 y=9$.
(b) $2 x-4 y=1$ $4 x-8 y=2$
(b) $2 x-4 y=1$ $4 x-8 y=2$

Ryo Kudo
Ryo Kudo
Numerade Educator
02:51

Problem 12

Under what conditions on $a$ and $b$ will the following linear system have no solutions, one solution, infinitely many solutions?
$$\begin{array}{l}2 x-3 y=a \\4 x-6 y=b\end{array}$$

Ryo Kudo
Ryo Kudo
Numerade Educator
09:30

Problem 13

Use parametric equations to describe the solution set of the linear equation.
(a) $7 x-5 y=3$
(b) $3 x_{1}-5 x_{2}+4 x_{3}=7$
(c) $-8 x_{1}+2 x_{2}-5 x_{3}+6 x_{4}=1$
(d) $3 v-8 w+2 x-y+4 z=0$

Ryo Kudo
Ryo Kudo
Numerade Educator
06:32

Problem 14

Use parametric equations to describe the solution set of the linear equation.
(a) $x+10 y=2$
(b) $x_{1}+3 x_{2}-12 x_{3}=3$
(c) $4 x_{1}+2 x_{2}+3 x_{3}+x_{4}=20$
(d) $v+w+x-5 y+7 z=0$

Ryo Kudo
Ryo Kudo
Numerade Educator
02:41

Problem 15

Each linear system has infinitely many solutions. Use parametric equations to describe its solution set.
(a) $2 x-3 y=1$ $6 x-9 y=3$
(b) $x_{1}+3 x_{2}-x_{3}=-4$ $3 x_{1}+9 x_{2}-3 x_{3}=-12$ $-x_{1}-3 x_{2}+x_{3}=4$

Destin Priester
Destin Priester
Numerade Educator
01:59

Problem 16

Each linear system has infinitely many solutions. Use parametric equations to describe its solution set.
(a) $6 x_{1}+2 x_{2}=-8$ $3 x_{1}+x_{2}=-4$
(b) $2 x-y+2 z=-4$ $6 x-3 y+6 z=-12$ $-4 x+2 y-4 z=8$

Destin Priester
Destin Priester
Numerade Educator
04:28

Problem 17

Find a single elementary row operation that will create a 1 in the upper left corner of the given augmented matrix and will not create any fractions in its first row.
(a) $\left[\begin{array}{rrrr}-3 & -1 & 2 & 4 \\ 2 & -3 & 3 & 2 \\ 0 & 2 & -3 & 1\end{array}\right]$
(b) $\left[\begin{array}{cccc}0 & -1 & -5 & 0 \\ 2 & -9 & 3 & 2 \\ 1 & 4 & -3 & 3\end{array}\right]$

Ryo Kudo
Ryo Kudo
Numerade Educator
05:15

Problem 18

Find a single elementary row operation that will create a 1 in the upper left corner of the given augmented matrix and will not create any fractions in its first row.
(a) $\left[\begin{array}{rrrr}2 & 4 & -6 & 8 \\ 7 & 1 & 4 & 3 \\ -5 & 4 & 2 & 7\end{array}\right] \quad$ (b) $\left[\begin{array}{rrrr}7 & -4 & -2 & 2 \\ 3 & -1 & 8 & 1 \\ -6 & 3 & -1 & 4\end{array}\right]$

Ryo Kudo
Ryo Kudo
Numerade Educator
03:19

Problem 19

Find all values of $k$ for which the given augmented matrix corresponds to a consistent linear system.
(a) $\left[\begin{array}{llr}1 & k & -4 \\ 4 & 8 & 2\end{array}\right]$
(b) $\left[\begin{array}{lll}1 & k & -1 \\ 4 & 8 & -4\end{array}\right]$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
05:45

Problem 20

Find all values of $k$ for which the given augmented matrix corresponds to a consistent linear system.
(a) $\left[\begin{array}{rrr}3 & -4 & k \\ -6 & 8 & 5\end{array}\right]$
(b) $\left[\begin{array}{rrr}k & 1 & -2 \\ 4 & -1 & 2\end{array}\right]$

Destin Priester
Destin Priester
Numerade Educator
04:38

Problem 21

The curve $y=a x^{2}+b x+c$ shown in the accompanying figure passes through the points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),$ and $\left(x_{3}, y_{3}\right)$ Show that the coefficients $a, b,$ and $c$ form a solution of the system of linear equations whose augmented matrix is $$\left[\begin{array}{llll} x_{1}^{2} & x_{1} & 1 & y_{1} \\ x_{2}^{2} & x_{2} & 1 & y_{2} \\ x_{3}^{2} & x_{3} & 1 & y_{3} \end{array}\right]$$ (FIGURE CAN'T COPY)

Ryo Kudo
Ryo Kudo
Numerade Educator
04:23

Problem 22

Explain why each of the three elementary row operations does not affect the solution set of a linear system.

Destin Priester
Destin Priester
Numerade Educator
02:45

Problem 23

Show that if the linear equations $$x_{1}+k x_{2}=c \quad \text { and } \quad x_{1}+l x_{2}=d$$
have the same solution set, then the two equations are identical (i.e., $k=l$ and $c=d$ ).

Destin Priester
Destin Priester
Numerade Educator
01:02

Problem 24

Consider the system of equations $$\begin{array}{l}a x+b y=k \\c x+d y=l \\e x+f y=m \end{array}$$ Discuss the relative positions of the lines $a x+b y=k$ $c x+d y=l,$ and $e x+f y=m$ when
(a) the system has no solutions.
(b) the system has exactly one solution.
(c) the system has infinitely many solutions.

Destin Priester
Destin Priester
Numerade Educator
01:33

Problem 25

Suppose that a certain diet calls for 7 units of fat, 9 units of protein, and 16 units of carbohydrates for the main meal, and suppose that an individual has three possible foods to choose from to meet these requirements:
Food 1: Each ounce contains 2 units of fat, 2 units of protein, and 4 units of carbohydrates. Food 2: Each ounce contains 3 units of fat, 1 unit of protein, and 2 units of carbohydrates. Food 3: Each ounce contains 1 unit of fat, 3 units of protein, and 5 units of carbohydrates.
Let $x, y,$ and $z$ denote the number of ounces of the first, second, and third foods that the dieter will consume at the main meal. Find (but do not solve) a linear system in $x, y,$ and $z$ whose solution tells how many ounces of each food must be consumed to meet the diet requirements.

Destin Priester
Destin Priester
Numerade Educator
01:48

Problem 26

Suppose that you want to find values for $a, b,$ and $c$ such that the parabola $y=a x^{2}+b x+c$ passes through the points $(1,1),(2,4),$ and $(-1,1) .$ Find (but do not solve) a system of linear equations whose solutions provide values for $a, b$ and $c .$ How many solutions would you expect this system of equations to have, and why?

Destin Priester
Destin Priester
Numerade Educator
01:38

Problem 27

Suppose you are asked to find three real numbers such that the sum of the numbers is $12,$ the sum of two times the first plus the second plus two times the third is $5,$ and the third number is one more than the first. Find (but do not solve) a linear system whose equations describe the three conditions.

Destin Priester
Destin Priester
Numerade Educator