1.1 Introduction to Systems of Linear Equations 1
CHAPTER 1: SYSTEMS OF LINEAR EQUATIONS AND MATRICES
1.1 Introduction to Systems of Linear Equations
1. (a) This is a linear equation in , , and .
(b) This is not a linear equation in , , and because of the term .
(c) We can rewrite this equation in the form therefore it is a linear equation in
, , and .
(d) This is not a linear equation in , , and because of the term .
(e) This is not a linear equation in , , and because of the term .
(f) This is a linear equation in , , and .
2. (a) This is a linear equation in and .
(b) This is not a linear equation in and because of the terms and .
(c) This is a linear equation in and .
(d) This is not a linear equation in and because of the term .
(e) This is not a linear equation in and because of the term .
(f) We can rewrite this equation in the form thus it is a linear equation in and .
3. (a)
(b)
(c)
4. (a) (b) (c)
,2 Chapter 1: Systems of Linear Equations and Matrices
5. (a) (b)
6. (a) (b)
7. (a) (b) (c)
8. (a) (b) (c)
9. The values in (a), (d), and (e) satisfy all three equations – these 3-tuples are solutions of the system.
The 3-tuples in (b) and (c) are not solutions of the system.
10. The values in (b), (d), and (e) satisfy all three equations – these 3-tuples are solutions of the system.
The 3-tuples in (a) and (c) are not solutions of the system.
11. (a) We can eliminate from the second equation by adding times the first equation to the
second. This yields the system
The second equation is contradictory, so the original system has no solutions. The lines
represented by the equations in that system have no points of intersection (the lines are
parallel and distinct).
(b) We can eliminate from the second equation by adding times the first equation to the
second. This yields the system
The second equation does not impose any restriction on and therefore we can omit it. The
lines represented by the original system have infinitely many points of intersection. Solving the
, 1.1 Introduction to Systems of Linear Equations 3
first equation for we obtain . This allows us to represent the solution using
parametric equations
where the parameter is an arbitrary real number.
(c) We can eliminate from the second equation by adding times the first equation to the
second. This yields the system
From the second equation we obtain . Substituting for into the first equation
results in . Therefore, the original system has the unique solution
The represented by the equations in that system have one point of intersection: .
12. We can eliminate from the second equation by adding times the first equation to the second.
This yields the system
If (i.e., ) then the second equation imposes no restriction on and ;
consequently, the system has infinitely many solutions.
If (i.e., ) then the second equation becomes contradictory thus the system has no
solutions.
There are no values of and for which the system has one solution.
13. (a) Solving the equation for we obtain therefore the solution set of the original
equation can be described by the parametric equations
where the parameter is an arbitrary real number.
(b) Solving the equation for we obtain therefore the solution set of the
original equation can be described by the parametric equations
where the parameters and are arbitrary real numbers.
(c) Solving the equation for we obtain therefore the solution set of
the original equation can be described by the parametric equations
, 4 Chapter 1: Systems of Linear Equations and Matrices
where the parameters , , and are arbitrary real numbers.
(d) Solving the equation for we obtain therefore the solution set of the
original equation can be described by the parametric equations
where the parameters , , , and are arbitrary real numbers.
14. (a) Solving the equation for we obtain therefore the solution set of the original
equation can be described by the parametric equations
where the parameter is an arbitrary real number.
(b) Solving the equation for we obtain therefore the solution set of the
original equation can be described by the parametric equations
where the parameters and are arbitrary real numbers.
(c) Solving the equation for we obtain therefore the solution set of
the original equation can be described by the parametric equations
where the parameters , , and are arbitrary real numbers.
(d) Solving the equation for we obtain therefore the solution set of the
original equation can be described by the parametric equations
where the parameters , , , and are arbitrary real numbers.
15. (a) We can eliminate from the second equation by adding times the first equation to the
second. This yields the system
The second equation does not impose any restriction on and therefore we can omit it.
Solving the first equation for we obtain . This allows us to represent the solution
using parametric equations
where the parameter is an arbitrary real number.
CHAPTER 1: SYSTEMS OF LINEAR EQUATIONS AND MATRICES
1.1 Introduction to Systems of Linear Equations
1. (a) This is a linear equation in , , and .
(b) This is not a linear equation in , , and because of the term .
(c) We can rewrite this equation in the form therefore it is a linear equation in
, , and .
(d) This is not a linear equation in , , and because of the term .
(e) This is not a linear equation in , , and because of the term .
(f) This is a linear equation in , , and .
2. (a) This is a linear equation in and .
(b) This is not a linear equation in and because of the terms and .
(c) This is a linear equation in and .
(d) This is not a linear equation in and because of the term .
(e) This is not a linear equation in and because of the term .
(f) We can rewrite this equation in the form thus it is a linear equation in and .
3. (a)
(b)
(c)
4. (a) (b) (c)
,2 Chapter 1: Systems of Linear Equations and Matrices
5. (a) (b)
6. (a) (b)
7. (a) (b) (c)
8. (a) (b) (c)
9. The values in (a), (d), and (e) satisfy all three equations – these 3-tuples are solutions of the system.
The 3-tuples in (b) and (c) are not solutions of the system.
10. The values in (b), (d), and (e) satisfy all three equations – these 3-tuples are solutions of the system.
The 3-tuples in (a) and (c) are not solutions of the system.
11. (a) We can eliminate from the second equation by adding times the first equation to the
second. This yields the system
The second equation is contradictory, so the original system has no solutions. The lines
represented by the equations in that system have no points of intersection (the lines are
parallel and distinct).
(b) We can eliminate from the second equation by adding times the first equation to the
second. This yields the system
The second equation does not impose any restriction on and therefore we can omit it. The
lines represented by the original system have infinitely many points of intersection. Solving the
, 1.1 Introduction to Systems of Linear Equations 3
first equation for we obtain . This allows us to represent the solution using
parametric equations
where the parameter is an arbitrary real number.
(c) We can eliminate from the second equation by adding times the first equation to the
second. This yields the system
From the second equation we obtain . Substituting for into the first equation
results in . Therefore, the original system has the unique solution
The represented by the equations in that system have one point of intersection: .
12. We can eliminate from the second equation by adding times the first equation to the second.
This yields the system
If (i.e., ) then the second equation imposes no restriction on and ;
consequently, the system has infinitely many solutions.
If (i.e., ) then the second equation becomes contradictory thus the system has no
solutions.
There are no values of and for which the system has one solution.
13. (a) Solving the equation for we obtain therefore the solution set of the original
equation can be described by the parametric equations
where the parameter is an arbitrary real number.
(b) Solving the equation for we obtain therefore the solution set of the
original equation can be described by the parametric equations
where the parameters and are arbitrary real numbers.
(c) Solving the equation for we obtain therefore the solution set of
the original equation can be described by the parametric equations
, 4 Chapter 1: Systems of Linear Equations and Matrices
where the parameters , , and are arbitrary real numbers.
(d) Solving the equation for we obtain therefore the solution set of the
original equation can be described by the parametric equations
where the parameters , , , and are arbitrary real numbers.
14. (a) Solving the equation for we obtain therefore the solution set of the original
equation can be described by the parametric equations
where the parameter is an arbitrary real number.
(b) Solving the equation for we obtain therefore the solution set of the
original equation can be described by the parametric equations
where the parameters and are arbitrary real numbers.
(c) Solving the equation for we obtain therefore the solution set of
the original equation can be described by the parametric equations
where the parameters , , and are arbitrary real numbers.
(d) Solving the equation for we obtain therefore the solution set of the
original equation can be described by the parametric equations
where the parameters , , , and are arbitrary real numbers.
15. (a) We can eliminate from the second equation by adding times the first equation to the
second. This yields the system
The second equation does not impose any restriction on and therefore we can omit it.
Solving the first equation for we obtain . This allows us to represent the solution
using parametric equations
where the parameter is an arbitrary real number.
