Download CHAPTER 4 Systems of Equations and Inequalities

CHAPTER 4
Systems of Equations and Inequalities
Section 4.1 (e-Book 6.1 & 6.2)
Linear and Nonlinear Systems of Equations in Two Variables
Definitions:
1. A linear system of equations in variables x and y has the general form:
,
(p)
are real numbers and x and y are the variables (unknowns).
where
2. If one or both equations in (p) are not linear then the system is called nonlinear.
Example 1:
a)
b)
c)
d)
e)
f)
By a solution of a system we mean an ordered pair of real numbers
which, upon
substitution, will satisfy all equations involved.
Example 2: For the system in example 1(b) above, show that the pairs
the pair
is a solution.
is not solution, but
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Solution Methods: There are several methods of solving a system of two equations. We will study
three such methods:
1. Substitution
2. Graphical
3. Elimination by Addition
(For linear systems only)
1. Substitution Method:
Step 1: Solve one of the two equations for x (or y), in terms of y (x).
Step 2: Substitute x (or y) in the other equation by the result from step 1 and simplify. This
new equation is now a simple equation in one unknown and can easily be solved for that
unknown.
Step 3: Now use the result of step 2 in step 1 to find the remaining unknown.
Example 3: Solve the following equations by substitution method:
a)
b)
c) A total of $3000 is put into two accounts that pay interest at 4% and 5%, respectively. In
5 years the total interest accumulated is $780. How much was put into each account?
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2. Graphical Method:
Step 1: Sketch the graphs of the two equations.
Step 2: The coordinates
of the intersection points of the two graphs (if any) are the
solutions of the system.
Remark 1: Clearly one of the following three situations will occur:
a) The two graphs intersect at one or more points which indicates that the system has one
or more solutions and. The system is called consistent and independent.
b) The two graphs do not intersect meaning that the system has no solutions. The system is
called inconsistent.
c) The two graphs coincide in which case the system has infinitely many solutions. The
system is called consistent and dependent.
Example 4: Solve the following systems by the graphing method:
a)
b)
c)
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3. Elimination by Addition (For linear equations only):
Step 1: Multiply each equation in (p) by some real numbers so that the coefficients of x (or
y) in both equations become the same with opposite signs.
Step 2: Add both equations, left side to left side and right side to side, and simplify. By
doing this x (or y) will be eliminated and the result is a linear equation in unknown y (or x)
and it can easily be solved for it.
Step 3. Repeat the process (or use substitution) to solve for the other unknown.
Example 5: Solve the following equations by elimination by addition method. Also sketch the
graphs.
a)
b)
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c) (No solution Case)
d) (Infinitely Many Solutions Case)
Example 6:
a) An algebra class has 31 students. The number of male students is 13 less than 3 time the
number of female students. Find the number of male and female students.
b) A store sells 20 chickens and 8 turkeys for $74. The store also sells 15 chickens and 24
turkeys for $87. Find the price of a chicken and price of a turkey.
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Exercise:
1.
Solve the following system by the graphing method:
a
2.
b)
Solve the following system by substitution method:
a)
b)
3.
Solve the following system by the method of elimination by addition:
4.
A manufacturer of office desks shipped 150 desks to its customers. The company makes a
profit of $25 per desk from sale to customer A and $20 to customer B. If the company’s total
profit is $3,325, how many desks were shipped to each customer?