Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Section 4.1: Systems of Equations
Systems of equations
A system of equations consists of two or more equations involving two or more variables
ax + by = c
dx + ey = f
A solution of such a system is an ordered pair (x, y) of real numbers that satisfies each equation in the
system. The solutions of the system correspond to the points of intersection of the graphs of the equations.
A system of linear equations can have exactly one solution, infinitely many solutions, or no solution.
Ex.1 Checking solutions of a system of equations.
Check whether each ordered pair is a solution of the system of equations
x+y =6
2x − 5y = −2
(1) (3, 3)
(2) (4, 2)
1
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Ex.2
Use the graphical method to solve the system of equations
2x + 3y = 7
2x − 5y = −1
The method of substitution
In order to solve a system of two equations involving two variables you need to follow these steps:
(1) Solve one of the equations for one variable in terms of the other.
(2) Substitute the expression obtained in Step 1 in the other equation to obtain an equation in one
variable.
(3) Solve the equation obtained in Step 2.
(4) Back-substitute the solution from Step 3 in the expression obtained in Step 1 to find the value of the
other variable.
(5) Check the solution to see that it satisfies both of the original equations.
2
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.3 Solve the following system of equations
−x + y = 3
3x + y = −1
Ex.4
Solve the following system of equations
2x − 2y = 0
x−y =1
3
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.5
Find two positive integers such that the sum of the numbers is 8 and their difference is 2.
4
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Section 4.2: Linear Systems in Two Variables
The method of elimination
The steps for solving a system of linear equations by the method of elimination are the following:
(1) Obtain coefficients for x (or y) that are opposites by multiplying all terms of one or both equations
by suitable constants.
(2) Add the equations to eliminate one variable, and solve the resulting equation.
(3) Back-substitute the value obtained in Step 2 in either the original equations and solve for the other
variable.
(4) Check your solution in both of the original equations.
Ex.1
Solve the following system of linear equations
3x + 2y = 4
5x − 2y = 8
5
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.2
Solve the following system of linear equations
Ex.3
Solve the following system of linear equations
4x − 5y = 13
3x − y = 7
3x + 9y = 8
2x + 6y = 7
6
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.4
Solve the following system of linear equations
−2x + 6y = 3
4x − 12y = −6
7
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Section 4.3: Linear Systems in Three Variables
Ex.1
Solve the following system of linear equations

 x − 2y + 2z
y + 2z

z
= 9
= 5
= 3
The method of Gauss elimination
Two systems of equations are equivalent if they have the same solution set. To solve a system that is not in
row-echelon form, first convert it to an equivalent system that is in row-echelon form and then solve the
new system.
Each of the following row operations on a system of linear equations produces an equivalent system of
linear equations.
(1) Interchange two equations.
(2) Multiply one of the equations by a nonzero constant.
(3) Add a multiple of one of the equations to another equation to replace the latter equation.
Ex.2
Solve the following system of linear equations
3x − 2y = −1
x−y =0
8
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.3
Solve the following system of linear equations

 x − 2y + 2z
−x + 3y

2x − 5y + z
9
= 9
= −4
= 10
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.4
Solve the following system of linear equations

 4x + y − 3z
2x − 3y + 2z

x+y+z
10
= 11
= 9
= −3
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.5
Solve the following system of linear equations

 x − 3y + z
2x − y − 2z

x + 2y − 3z
11
= 1
= 2
= −1
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.6
Solve the following system of linear equations

 x + y − 3z
y−z

−x + 2y
= −1
= 0
= 1
Solutions of a linear system
For a system of linear equations, exactly one of the following is true.
(1) There is exactly one solution.
(2) There are infinitely many solutions.
(3) There is no solution.
12
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Section 4.4: Matrices and Linear Systems
Matrices
A matrix (plural matrices) is a rectangular array of real numbers, such as


3 1 −1
 2 5 0 
2 4 1
An item in a matrix is called an entry or an element. If the matrix has n rows and m columns, its order is
n × m. A matrix with the same number of rows and columns is called a square matrix.
Ex.1
Determine the order of each matrix.
(1)
1
0
2
5
4
−3
(2)
0
0
0
0
(3)

1
 5
4

−1
0 
−2
Augmented matrices
A matrix derived from a system of linear equations is the augmented matrix. The matrix derived from the
coefficients of the system (but not including the constant terms) is the coefficient matrix of the system.
13
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.2
Form the coefficient matrix and the augmented matrix for each system.
(1)
−x + 5y = 2
7x − 2y = −6
(2)

 3x + 2y − z = 1
x + 2z = −3

−2x − y = 4
Ex.3
Write the system of linear equations represented by each matrix.
(1)


..
 3 −5 . 4 
.
−1 2 .. 0
(2)


..
1
3
.
2


..
0 1 . −3
(3)


..
2
0
−8
.
1




..
 −1 1
1 . 2 


.
5 −1 7 .. 3
14
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Elementary row operations
As in the case of systems of linear equations there are three elementary row operations that produce equivalent
matrices. The elementary row operations are
(1) Interchange two rows.
(2) Multiply a row by a nonzero constant.
(3) Add a multiple of a row to another row.
Ex.4
Given the matrix
1
0
2
5
do the following elementary row operations
(1) Interchange the first and second row.
(2) Multiply the first row by 3.
(3) Add −2 times the first row to the second row.
15
4
−3
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Definition of row-echelon form of a matrix
A matrix is in row-echelon form if
(1) All entries at the bottom of the diagonal are equal to zero.
(2) The first nonzero entry in a row is 1 (leading 1).
(3) For two consecutive (nonzero) rows, the leading 1 in the higher row is farther to the left than the
leading 1 in the lower row.
Ex.5
The matrix

1
 0
0
2
1
0

4
−3 
1
is in row-echelon form.
Gaussian elimination
We can perform the Gaussian elimination using matrices and the following steps:
(1) Write the augmented matrix of the system of linear equations.
(2) Use elementary row operations to rewrite the augmented matrix in row-echelon form.
(3) Write the system of linear equations corresponding to the matrix in row-echelon form, and use
back-substitution to find the solution.
16
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.6
Solve the following system of linear equations
2x − 3y
x + 2y
17
= −2
= 13
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.7
Solve the following system of linear equations

 3x + 3y
2x − 3z

6y + 4z
18
= 9
= 10
= −12
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.8
Solve the following system of linear equations
6x − 10y
9x − 15y
19
=
=
Math 1010
−4
5
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.9
Solve the following system of linear equations
12x − 6y
−8x + 4y
20
= −3
= 2
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Section 4.5: Determinants and Linear Systems
Determinant of a 2 × 2 matrix
Given a matrix 2 × 2
a1
a2
A=
the determinant is
a1
det(A) = a2
b1
b2
b1 = a1 b2 − a2 b1
b2 Ex.1
Find the determinant of each matrix.
(1)
−3
4
2
1
(2)
−1
2
2
−4
(3)
1
2
21
3
5
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Determinant of a 3 × 3 matrix
Given a matrix 3 × 3

a1
A =  a2
a3
the determinant is
b2
det(A) = a1 b3
b1
b2
b3
c2 − b1 a2
a3
c3 Math 1010

c1
c2 
c3
c2 + c1 a2
a3
c3 Ex.2
Find the determinant of each matrix.
(1)

1 2
2 3 
4 2

−1
 0
3
(2)

1
 3
4
2
0
0
22

1
2 
−1
b2 b3 Chapter 4: Systems of Equations and Ineq.
Lecture notes
Math 1010
Cramer’s rule
For the system of linear equations
a1 x + b1 y
a2 x + b2 y
=
=
c1
c2
the solution is given by
Dx
x=
=
D
c1
c2
a1
a2
b1
b2
b1
b2
Dy
y=
=
D
a1
a2
a1
a2
c1
c2
b1
b2

 a1 x + b1 y + c1 z
a2 x + b2 y + c2 z

a3 x + b3 y + c3 z
=
=
=
where D 6= 0.
For the system of linear equations
d1
d2
d3
the solution is given by
Dx
x=
=
D
d1
d2
d3
a1
a2
a3
b1
b2
b3
b1
b2
b3
c1
c2
c3
c1
c2
c3
Dy
= y=
D
a1
a2
a3
a1
a2
a3
d1
d2
d3
b1
b2
b3
c1
c2
c3
c1
c2
c3
Dz
= z=
D
a1
a2
a3
a1
a2
a3
b1
b2
b3
b1
b2
b3
d1
d2
d3
c1
c2
c3
where D 6= 0.
23
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.3
Use Cramer’s rule to solve the following system of linear equations
4x − 2y = 10
3x − 5y = 11
24
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.4
Use Cramer’s rule to solve the following system of linear equations

 −x + 2y − 3z = 1
2x + z = 0

3x − 4y + 4z = 2
25
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Section 4.6: Systems of Linear Inequalities
Graphing a system of linear inequalities
(1) Sketch the line that corresponds to each inequality.
(2) Lightly shade the half-plane that is the graph of each linear inequality.
(3) The graph of the system is the intersection of the half-planes.
Ex.1
Sketch the graph of the following system of linear inequalities
2x − y ≤ 5
x + 2y ≥ 2
26
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.2
Sketch the graph of the following system of linear inequalities
y < 4
y > 1
27
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.3
Sketch the graph of the following system of linear inequalities

 x−y < 2
x > −2

y ≤ 3
Ex.4
Sketch the graph of the following system of linear inequalities

x+y ≤ 5



3x + 2y ≤ 12
x ≥ 0



y ≥ 0
28
Math 1010
Chapter 4: Systems of Equations and Ineq.
Lecture notes
Ex.5
Find a system of inequalities that defines the region shown below.
29
Math 1010