Download Chapter 4 - Systems of Equations and Inequalities

Math 233 - Spring 2009
Chapter 4 - Systems of Equations and Inequalities
4.1
Solving Systems of equations in Two Variables
Definition 1. A system of linear equations is two or more linear equations to which we try to
find a common solution.
EX 1. Consider the equations:
y
= x+4
y
=
2x + 1
A solution to a system of linear equtions is an ordered pair or pairs that satisfy ALL equations
in the system. Let’s verify that (3, 7) is a solution to the given system.
4.1.1
Solve Systems of Linear Equations Graphically
To solve a system with two equations in two variables we graph them both on the same axes and
common solution is the point where they intersect.
EX 2. 1. Solve the following system of equations graphically.
y
= x+3
y
= −x + 1
6
-
2. Let’s consider all the possible ways that two lines can interact:
6
6
-
(a)
6
-
(b)
-
(c)
1
4.1.2
Solving Systems by Substitution
It is usually not practical to solve systems by graphing (messy drawings, points may not be exactly
clear) so we need more precies methods which we now develop. Our first algebraic method is called
substitution.
Substitution Method
1. Pick an equation and solve for one of the variables. (Note: It is usually easiest to solve for a
variable with coefficient of 1.)
2. Substitute the expression found for the variable in step 1 into the other equation. We
now have an equation with only one variable.
3. Solve the equation in step 2 to find the value of one of the variables.
4. Substitute the value found in step 3 into the equation from step 1. Solve the equation to find
the other variable.
5. Check your solution in all of the equations in the system.
EX 3. 1. Solve using the substitution method.
2x − 10
y
=
y
= −3x + 5
2. Solve using the substitution method.
4.1.3
3x + y
=
−1
x + 2y
=
−12
Solving Systems Using the Addition Method
Often the easiest method for solving a system of equations is the addition method (also known as
the elimination method). Before we list the steps, let’s look at an example.
2
EX 4. Solve the following system of equations using the addition method.
x + 4y
=
9
3x − 4y
=
11
Addition Method
1. If necessary, rewrite each equation in standard form.
2. If necessary, multiply one or both equations by a constant so that when the equations are
added, the sum will contain only one variable.
3. Add the equations. We get an equation in one variable.
4. Solve the equation found in step 3.
5. Substitute the value found in step 4 into either of the original equations and solve for the other
variable.
6. Check your solution in all of the equations in the system.
EX 5. 1. Solve using the addition method:
3x + y
= −1
x + 2y
=
−12
2. Solve using the addition method:
3x + 2y
=
5
2x + 3y
=
4
3
3. Solve using the addition method:
x =
2
1
x+
=
2
3
13 − 5y
1
4. Solve using the addition method:
x + 4y
−3x − 12y
4.2
= −7
= −17
Solving Systems of Linear Equations in Three Variables
An equation of the form 3x − 2y + 4z = 11 is a linear equation in three variables. The solution
to such an equation is an ordered triple of the form (x, y, z). For, example a solution of the above
equation is (1, 2, 3). We will consider systems of three equations in three variables.
To solve systems of equations in three variables we will use the methods we have already introduced, namely the substitution and addition methods.
EX 6. 1. Solve the following system using the substitution method.
2x
=
8
2x + y
=
3
3x + 5y + 6z
4
= −1
2. Solve the following system using the addition method.
2x
x
+
3y
+
-
y
4z
2z
3z
=
=
=
-4
12
19
=
=
=
-4
10
7
3. Solve the following system using the addition method.
-2x
7x
6x
4.3
+
+
-
3y
4y
9y
+
+
-
z
2z
3z
Applications and Problem Solving
There are countless applications for systems of equations. We will examine a couple examples.
EX 7. 1. How many pints of a 10% salt solution and a 50% salt solution must be combined to get 44
pints of a 40% salt solution?
5
2. On an algebra test, a total of 75 students had grades of A or B. There were 5 more students
with A’s than B’s. Find the number of students who had an A and the number of students
who had a B.
3. Three brothers invest a total of $35,000 a business. The first brother earns 3% profit on his
investment. The second brother earns a 5% profit on his investment. The third brother earns
7% profit on his investment. The third brother invested $5000 more than the sum of the other
two brothers. After one year, the profit earned was $2050. How much did each brother invest?
4.4
4.4.1
Solving Systems of Equations Using Matrices
Matrices
A matrix is a rectangular array of numbers within brackets. The plural of matrix is matrices.
EX 8. Here are a couple matrices:
1
4
2
5
3
6
or
1
3
2
4
The numbers inside the matrices are referred to as elements. We refer to the size of the matrix
by the number of rows and columns, thus the left matrix has 2 rows and 3 columns so it is a 2 × 3
matrix. A square matrix is a matrix which has the same number of rows as columns.
We will use matrices to solve our systems of equations. In order to do this we will represent our
system with an augmented matrix. Consider the system of equations
a1 x + b1 y
= c1
a2 x + b2 y
= c2
the corresponding matrix is written:
a1
a2
b1
b2
6
c1
c2
EX 9. Express the following system of linear equations with an augmented matrix.
4.4.2
3x − 2y
=
9
12x + 5y
=
10
Solve Systems of Linear Equations
In order to solve systems of equations using matrices we will attempt to transform our matrices using
3 simple transformations. Our goal is to rewrite the matrix in the following form (called triangular
form):
1 a p
0 1 q
To transform a matrix into this form we have the following row transformations:
Procedures for Row Transformations
1. Any row may be multiplied (or divided) by a nonzero number.
2. All the numbers in a row may be multiplied by any nonzero real number, then added to any
other row.
3. The order of the rows may be switched.
EX 10. 1. Solve the following system of equations using matrices.
3x − 2y
=
9
12x + 5y
=
10
7
2. Solve the following system of equations using matrices.
4x + 3y
=
−2
x + 3y
=
4
3. Solve the following system of equations using matrices.
2x − 3y
−4x + 9y
4.4.3
=
3
= −7
Recognize Inconsistent and Dependent Systems
Recall that we call a system of equations inconsistent if there are no common solution to the
equations of the system (parallel lines). A dependent system is one with an infinite number of
solutions.
Let’s examine how to determine, using matrices, if a system is inconsistent or dependent.
EX 11. 1. Solve the system of equations using matrices.
−2x − 4y
3x + 6y
8
=
7
= −8
2. Solve the system of equations using matrices.
4.5
4.5.1
x − 3y
=
−4
2x − 6y
=
−8
Determinants and Cramer’s Rule
Evaluate a Determinant of a 2 × 2 Matrix
For every 2 × 2 matrix we can find a number
associated
withit called
a b
a1 b1
The determinant of a 2 × 2 matrix
is denoted 1 1
a2 b2
a2 b2
a1 b1 a2 b2 = a1 b2 − a2 b1
the
determinant.
and is evaluated as follows:
EX 12. Evaluate the following determinants:
3 −2 1. 1 −7 5
2. −2
4.5.2
4 3 Cramer’s Rule
We will study now a cool rule that gives us a formula for solving systems of linear equations. We
will only study this for systems with two variables but it works for three or more variable system of
equations. First some notation.
Given the system of equations:
a1 x + b1 y
= c1
a 2 x + b2 y
= c2
We denote by D the following:
a
D = 1
a2
b1 = a1 b2 − a2 b1
b2 Further we denote with Dx and Dy :
c
Dx = 1
c2
b1 = c1 b2 − c2 b1
b2 9
and
a
Dy = 1
a2
c1 = a1 c2 − a2 c1
c2 Finally, we have Cramer’s Rule: For the system of equations
a1 x + b1 y
= c1
a2 x + b2 y
= c2
Then the solution is given by
and
Dx
x=
= D
c1
c2
a1
a2
b1
b2
b1
b2
Dy
= y=
D
a1
a2
a1
a2
c1
c2
b1
b2
EX 13. Solve the system of equations using Cramer’s rule.
3x + 5y
=
9
4x − y
=
−11
REMARK 1. When D = 0:
• If D = 0, Dx = 0, and Dy = 0, then the system is dependent.
• If D = 0 and either Dx 6= 0 or Dy 6= 0, then the system is inconsistent.
4.6
Solving Systems of Linear Inequalities
Recall, we learned how to graph linear inequalities in two variables:
EX 14. Graph the inequality 2x + 3y > 6
6
-
We now investigate how to solve systems of linear inequalities graphically.
To do this, we graph each inequality on the same axes. The solution is the set of points whose
coordinates satisfy all the inequalities in the system.
10
EX 15. 1. Graph:
y
x−y
1
− x+6
3
< 2
≤
6
-
2. Graph
2x − y
≤ 4
3x + 2y
> 6
6
-
3. Graph
x ≥ 0
y
≥ 0
x+y
≤ 5
3x + y
≤ 9
6
-
11