Precalculus

Carl Stitz, Jeff Zeager

Chapter 8

Systems of Equations and Matrices - all with Video Answers

Educators

Section 1

Systems of Linear Equations: Gaussian Elimination

Problem 1

In Exercises $1-8$, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$
\left\{\begin{array}{r}
x+2 y=5 \\
x=6
\end{array}\right.
$$

Erika Bustos

Erika Bustos

Numerade Educator

Problem 2

In Exercises $1-8$, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$
\left\{\begin{aligned}
2 y-3 x &=1 \\
y &=-3
\end{aligned}\right.
$$

Erika Bustos

Numerade Educator

Problem 3

In Exercises $1-8$, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$
\left\{\begin{array}{l}
\frac{x+2 y}{4}=-5 \\
\frac{3 x-y}{2}=1
\end{array}\right.
$$

Erika Bustos

Numerade Educator

Problem 4

In Exercises $1-8$, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$
\left\{\begin{array}{l}
\frac{2}{3} x-\frac{1}{5} y=3 \\
\frac{1}{2} x+\frac{3}{4} y=1
\end{array}\right.
$$

Erika Bustos

Numerade Educator

Problem 5

In Exercises $1-8$, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$
\left\{\begin{aligned}
\frac{1}{2} x-\frac{1}{3} y &=-1 \\
2 y-3 x &=6
\end{aligned}\right.
$$

Erika Bustos

Numerade Educator

Problem 6

In Exercises $1-8$, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$
\left\{\begin{array}{r}
x+4 y=6 \\
\frac{1}{12} x+\frac{1}{3} y=\frac{1}{2}
\end{array}\right.
$$

Erika Bustos

Numerade Educator

Problem 7

In Exercises $1-8$, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$
\left\{\begin{aligned}
3 y-\frac{3}{2} x &=-\frac{15}{2} \\
\frac{1}{2} x-y &=\frac{3}{2}
\end{aligned}\right.
$$

Erika Bustos

Numerade Educator

Problem 8

In Exercises $1-8$, take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically.
$$
\left\{\begin{aligned}
\frac{5}{6} x+\frac{5}{3} y &=-\frac{7}{3} \\
-\frac{10}{3} x-\frac{20}{3} y &=10
\end{aligned}\right.
$$

Erika Bustos

Numerade Educator

Problem 9

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{array}{rr}
-5 x+y= & 17 \\
x+y= & 5
\end{array}\right.
$$

Erika Bustos

Numerade Educator

Problem 10

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
x+y+z &=3 \\
2 x-y+z &=0 \\
-3 x+5 y+7 z &=7
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 11

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
4 x-y+z &=5 \\
2 y+6 z &=30 \\
x+z &=5
\end{aligned}\right.
$$

Erika Bustos

Numerade Educator

Problem 12

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
4 x-y+z &=5 \\
2 y+6 z &=30 \\
x+z &=6
\end{aligned}\right.
$$

Erika Bustos

Numerade Educator

Problem 13

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{array}{rr}
x+y+z= & -17 \\
y-3 z= & 0
\end{array}\right.
$$

James Kiss

Numerade Educator

Problem 14

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
x-2 y+3 z &=7 \\
-3 x+y+2 z &=-5 \\
2 x+2 y+z &=3
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 15

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{array}{rr}
3 x-2 y+z= & -5 \\
x+3 y-z= & 12 \\
x+y+2 z= & 0
\end{array}\right.
$$

James Kiss

Numerade Educator

Problem 16

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
2 x-y+z &=-1 \\
4 x+3 y+5 z &=1 \\
5 y+3 z &=4
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 17

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
x-y+z &=-4 \\
-3 x+2 y+4 z &=-5 \\
x-5 y+2 z &=-18
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 18

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
2 x-4 y+z &=-7 \\
x-2 y+2 z &=-2 \\
-x+4 y-2 z &=3
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 19

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
2 x-y+z &=1 \\
2 x+2 y-z &=1 \\
3 x+6 y+4 z &=9
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 20

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
x-3 y-4 z &=3 \\
3 x+4 y-z &=13 \\
2 x-19 y-19 z &=2
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 21

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
x+y+z &=4 \\
2 x-4 y-z &=-1 \\
x-y &=2
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 22

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
x-y+z &=8 \\
3 x+3 y-9 z &=-6 \\
7 x-2 y+5 z &=39
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 23

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
2 x-3 y+z &=-1 \\
4 x-4 y+4 z &=-13 \\
6 x-5 y+7 z &=-25
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 24

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
2 x_{1}+x_{2}-12 x_{3}-x_{4} &=16 \\
-x_{1}+x_{2}+12 x_{3}-4 x_{4} &=-5 \\
3 x_{1}+2 x_{2}-16 x_{3}-3 x_{4} &=25 \\
x_{1}+2 x_{2}-5 x_{4} &=11
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 25

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
x_{1}-x_{3} &=-2 \\
2 x_{2}-x_{4} &=0 \\
x_{1}-2 x_{2}+x_{3} &=0 \\
-x_{3}+x_{4} &=1
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 26

In Exercises $9-26$, put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$
\left\{\begin{aligned}
x_{1}-x_{2}-5 x_{3}+3 x_{4} &=-1 \\
x_{1}+x_{2}+5 x_{3}-3 x_{4} &=0 \\
x_{2}+5 x_{3}-3 x_{4} &=1 \\
x_{1}-2 x_{2}-10 x_{3}+6 x_{4} &=-1
\end{aligned}\right.
$$

James Kiss

Numerade Educator

Problem 27

Find two other forms of the parametric solution to Exercise 11 above by reorganizing the equations so that $x$ or $y$ can be the free variable.

Samuel Hannah

Numerade Educator

Problem 28

A local buffet charges $\$ 7.50$ per person for the basie buffet and $\$ 9.25$ for the deluxe buffet (which includes crab legs.) If 27 diners went out to eat and the total bill was $\$ 227.00$ before taxes, how many chose the basic buffet and how many chose the deluxe buffet?

Erika Bustos

Numerade Educator

Problem 29

At The Old Home Fill'er Up and Keep on a-Truckin' Cafe, Mavis mixes two different types of coffee beans to produce a house blend. The first type costs $\$ 3$ per pound and the second costs $\$ 8$ per pound. How much of each type does Mavis use to make 50 pounds of a blend which costs $\$ 6$ per pound?

Debasish Das

Numerade Educator

Problem 30

Skippy has a total of $\$ 10,000$ to split between two investments, One account offers $3 \%$ simple interest, and the other account offers $8 \%$ simple interest. For tax reasons, he can only earn $\$ 500$ in interest the entire year. How much money should Skippy invest in each account to earn $\$ 500$ in interest for the year?

Erika Bustos

Numerade Educator

Problem 31

A $10 \%$ salt solution is to be mixed with pure water to produce 75 gallons of a $3 \%$ salt solution. How much of each are needed?

BM

Barbara Manley

Numerade Educator

Problem 32

At The Crispy Critter's Head Shop and Patchouli Emporium along with their dried up weeds, sunflower seeds and astrologieal posteards they sell an herbal tea blend. By weight, Type I herbal tea is $30 \%$ peppermint, $40 \%$ rose hips and $30 \%$ chamomile, Type II has percents $40 \%$, $20 \%$ and $40 \%$, respectively, and Type III has percents $35 \%, 30 \%$ and $35 \%$, respectively. How much of each Type of tea is needed to make 2 pounds of a new blend of tea that is equal parts peppermint, rose hips and chamomile?

Yujie Wang

College of San Mateo

Problem 33

Discuss with your classmates how you would approach Exercise 32 above if they needed to use up a pound of Type I tea to make room on the shelf for a new canister.

AG

Ankit Gupta

Numerade Educator

Problem 34

If you were to try to make $100 \mathrm{~mL}$ of a $60 \%$ acid solution using stock solutions at $20 \%$ and $40 \%$, respectively, what would the triangular form of the resulting system look like? Explain.

AG

Ankit Gupta

Numerade Educator