Unformatted text preview: MAC 1105
Module 3
System of Equations and
Inequalities
Rev.S08 Learning Objectives
Upon completing this module, you should be able to:
1.
2.
3.
4.
5.
6. Evaluate functions of two variables.
Apply the method of substitution.
Apply the elimination method.
Solve system of equations symbolically.
Apply graphical and numerical methods to system of
equations.
Recognize different types of linear systems. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 2 Learning Objectives (Cont.)
7.
8.
9.
10.
11.
12. Rev.S08 Use basic terminology related to inequalities.
Use interval notation.
Solve linear inequalities symbolically.
Solve linear inequalities graphically and numerically.
Solve double inequalities.
Graph a system of linear inequalities. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 3 System of Equations and Inequalities
There are two major topics in this module:  System of Linear Equations in Two Variables
System
 Solutions of Linear Inequalities Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 4 Do We Really Use Functions of Two
Variables?
The answer is YES. Many quantities in everyday life depend on more than
one variable.
Examples Area of a rectangle requires both width and length. Heat index is the function of temperature and humidity. Wind chill is determined by calculating the temperature
and wind speed. Grade point average is computed using grades and
credit hours. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 5 Let’s Take a Look at the
Arithmetic Operations
The arithmetic operations of addition, subtraction,
multiplication, and division are computed by functions
of two inputs. The addition function of f can be represented
symbolically by f(x,y) = x + y, where z = f(x,y). The independent variables are x and y. The dependent variable is z. The z output depends
on the inputs x and y. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 6 Here are Some Examples
For each function, evaluate the expression and interpret
the result.
a) f(5, –2) where f(x,y) = xy
b) A(6,9), where
calculates the area of a
triangle with a base of 6 inches and a height of 9
inches.
Solution
•
f(5, –2) = (5)(–2) = –10.
• A(6,9) =
If a triangle has a base of 6 inches and a height of 9
inches, the area of the triangle is 27 square inches.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 7 What is a System of Linear Equations?
A linear equation in two variables can be written in the
form ax + by = k, where a, b, and k are constants, and
a and b are not equal to 0. A pair of equations is called a system of linear
equations because they involve solving more than one
linear equation at once. A solution to a system of equations consists of an xvalue and a yvalue that satisfy both equations
simultaneously. The set of all solutions is called the solution set. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 8 How to Use the Method of Substitution to
solve a system of two equations? Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 9 How to Solve the System Symbolically?
Solve the system symbolically.
Solution
Step 1: Solve one of the
equations for one of the
variables. Rev.S08 Step 2: Substitute
for y in the second
equation. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 10 How to Solve the System Symbolically?
(Cont.)
Step 3: Substitute x = 1 into the equation
from Step 1. We find that
Check: The ordered pair is (1, 2) since the solutions check in
both equations. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 11 Example with Infinitely Many Solutions
Solve the system.
• Solution
• Solve the second equation for y. • Substitute 4x + 2 for y in the first equation, solving for
x. • The equation − 4 = − 4 is an identity that is always
true and indicates that there are infinitely many
solutions. The two equations are equivalent.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 12 Possible Graphs of a System of Two
Linear Equations in Two Variables Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 13 How to Use Elimination Method to Solve
System of Equations?
Use elimination to solve each system of equations, if
possible. Identify the system as consistent or
inconsistent. If the system is consistent, state whether
the equations are dependent or independent. Support
your results graphically.
a) 3x − y = 7
5x + y = 9 Rev.S08 b) 5x − y = 8
c) x − y = 5
− 5x + y = − 8
x− y=− 2 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 14 How to Use Elimination Method to Solve
System of Equations? (Cont.) Solution
a) Eliminate y by adding
Eliminate by
the equations. Find y by substituting
x = 2 in either equation. The solution is (2, − 1). The system is
consistent and the equations are independent.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 15 How to Use Elimination Method to Solve
System of Equations? (Cont.)
b) If we add the equations we obtain the
If
following result.
following The equation 0 = 0 is an
The
identity that is always true.
always
The two equations are equivalent.
There are infinitely
many solutions.
many
{(x, y) 5x − y = 8}
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 16 How to Use Elimination Method to Solve
System of Equations? (Cont.)
c) If we subtract the second equation from
the first, we obtain the following result. The equation 0 = 7 is a
The
contradiction that is never true.
never
Therefore there are no solutions,
no
and the system is inconsistent.
system
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 17 Let’s Practice Using Elimination
Solve the system by using elimination. Solution
Multiply the first equation by 3 and the second equation
by 4. Addition eliminates the yvariable. Substituting x = 3 in 2x + 3y = 12 results in
2(3) + 3y = 12 or y = 2
The solution is (3, 2).
http://faculty.valenciacc.edu/ashaw/
Rev.S08 Click link to download other modules. 18 Terminology related to Inequalities
• Inequalities result whenever the equals sign in
Inequalities
equals
an equation is replaced with any one of the
replaced
symbols: ≤, ≥, <, >
≤,
• Examples of inequalities include:
Examples
•2x –7 > x +13
2x
•x2 ≤ 15 – 21x
•xy +9 x < 2x2
xy
•35 > 6
35
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 19 Linear Inequality in One Variable
•A linear inequality in one variable is an inequality that can
linear
be written in the form
ax + b > 0 where a ≠ 0.
ax
(The symbol may be replaced by ≤, ≥, <, > )
≤,
•Examples of linear inequalities in one variable:
• 5x + 4 ≤ 2 + 3x simplifies to 2x + 2 ≤ 0
• − 1(x – 3) + 4(2x + 1) > 5 simplifies to 7x + 2 > 0
•Examples of inequalities in one variable which are not
Examples
linear:
linear:
• x2 < 1
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 20 Let’s Look at Interval Notation The solution to a linear inequality in one variable is typically an
solution
linear
interval on the real number line. See examples of interval notation
interval
See
below.
below. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 21 Multiplied by a Negative Number
Note that 3 < 5, but if both sides are multiplied by − 1, iin
n
order to produce a true statement the > symbol must be
used.
used
3<5
but
but
− 3>− 5
So when both sides of an inequality are multiplied (or
So
divided) by a negative number the direction of the
inequality must be reversed.
inequality Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 22 How to Solve Linear Inequalities
Symbolically? The procedure for solving a linear inequality symbolically is the same as
the procedure for solving a linear equation, except when both sides
except
of an inequality are multiplied (or divided) by a negative number
the direction of the inequality is reversed.
Example of Solving a
Linear Equation Symbolically
Linear Equation
Solve − 2x + 1 = x − 2
Solve
− 2x − x = − 2 − 1
− 3x = − 3
x=1 Rev.S08 Example of Solving a
Example
Linear Inequality Symboliclly
Linear Inequality
Solve − 2x + 1 < x − 2
Solve
− 2x − x < − 2 − 1
− 3x < − 3
x>1
Note that we divided both
Note
sides by − 3 so the direction
so
of the inequality was
reversed. In interval notation
the solution set is (1,∞).
the http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 23 How to Solve a Linear Inequality
Graphically?
Solve Note that the graphs intersect at the point (8.20, 7.59). The graph
of y1 is above the graph of y2 to the right of the point of intersection
o f is
or when x > 8.20. Thus, in interval notation, the solution set is
interval
(8.20, ∞)
(8.20, Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 24 How to Solve a Linear Inequality
Numerically?
Solve Note that the inequality above becomes y1 ≥ y2 since we let y1 equal the leftequal
hand side and y2 equal the right hand side.
hand
To write the solution set of the inequality we are looking for the values of x in
the table for which y1 is the same or larger than y2. Note that when x = − 1.3, y1
1.3,
is less than y2; but when x = − 1.4, y1 is larger than y2. By the Intermediate
Value Property, there is a value of x between − 1.4 and − 1.3 such that y1 = y2.
In order to find an approximation of this value, make a new table in which x is
incremented by .01 (note that x is incremented by .1 in the table to the left
here.)
here.)
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 25 How to Solve a Linear Inequality
Numerically? (cont.)
Solve
Solve To write the solution set of the inequality we are looking for the values
To
solution
of x in the table for which y1 is the same as or larger than y2. Note that
when x is approximately − 1.36, y1 equals y2 and when x is smaller
than − 1.36 y1 is larger than y2 , so the solutions can be written
1.36
so
x ≤ − 1.36 or (− ∞, − 1.36] in interval notation.
∞, Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 26 How to Solve Double Inequalities?
• Example: Suppose the Fahrenheit temperature x
miles above the ground level is given by
T(x) = 88 – 32 x. Determine the altitudes where the
88
Determine
air temp is from 300 to 400.
air • We must solve the inequality
30 < 88 – 32 x < 40
40 To solve: Isolate the variable x in the middle of the
To
Isolate
the
threepart inequality
threepart
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 27 How to Solve Double Inequalities?
(Cont.) Direction reversed –Divided each
side of an inequality by a negative
side
Thus, between 1.5 and 1.8215
Thus,
miles above ground level,
the air temperature is
between 30 and 40 degrees
Fahrenheit.
Fahrenheit.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 28 How to Graph a System of Linear
Inequalities?
The graph of a linear inequality is a halfplane, which
may include the boundary. The boundary line is included
when the inequality includes a less than or equal to or
greater than or equal to symbol.
To determine which part of the plane to shade, select a
test point. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 29 How to Graph a System of Linear
Inequalities? (Cont.)
Graph the solution set to the inequality x + 4y > 4.
Solution
Graph the line x + 4y = 4 using a dashed line.
Use a test point to determine which half of the plane to
shade.
Test
Point x + 4y > 4 True or
False? (4, 2) 4 + 4(2) > 4 True (0, 0) 0 + 4(0) > 4 False Rev.S08
http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 30 Example
Solve the system of inequalities
by shading the solution set. Use
the graph to identify one solution.
x+y≤3
2x + y ≥ 4
Solution
Solve each inequality for y.
y ≤ − x + 3 (shade below line)
y ≥ − 2x + 4 (shade above line)
The point (4, − 2) is a solution.
Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 31 What have we learned?
We have learned to:
1.
2.
3.
4.
5.
6. Evaluate functions of two variables.
Apply the method of substitution.
Apply the elimination method.
Solve system of equations symbolically.
Apply graphical and numerical methods to system of
equations.
Recognize different types of linear systems. Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 32 What have we learned? (Cont.)
7.
8.
9.
10.
11.
12. Rev.S08 Use basic terminology related to inequalities.
Use interval notation.
Solve linear inequalities symbolically.
Solve linear inequalities graphically and numerically.
Solve double inequalities.
Graph a system of linear inequalities. http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 33 Credit
Some of these slides have been adapted/modified in part/whole from the
slides of the following textbook:
• Rockswold, Gary, Precalculus with Modeling and Visualization, 3th
Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/
Click link to download other modules. 34 ...
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Full Document
 Winter '08
 Staff
 Algebra, Equations, Inequalities, Elementary algebra, Click Link, Solve Linear Inequalities, Solutions of Linear Inequalities

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