Transcript in Equation 1

7
Systems of Equations
and Inequalities
Copyright © Cengage Learning. All rights reserved.
7.1
Linear and Nonlinear
Systems of Equations
Copyright © Cengage Learning. All rights reserved.
Objectives
 Use the method of substitution to solve systems
of linear equations in two variables.
 Use the method of substitution to solve systems
of nonlinear equations in two variables.
 Use a graphical approach to solve systems of
equations in two variables.
 Use systems of equations to model and solve
real-life problems.
3
The Method of Substitution
4
The Method of Substitution
Most problems have involved either a function of one
variable or a single equation in two variables.
However, many problems in science, business, and
engineering involve two or more equations in two or more
variables.
To solve such a problem, you need to find solutions of a
system of equations.
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The Method of Substitution
Here is an example of a system of two equations in two
unknowns.
2x + y = 5
Equation 1
3x – 2y = 4
Equation 2
A solution of this system is an ordered pair that satisfies
each equation in the system.
Finding the set of all solutions is called solving the system
of equations.
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The Method of Substitution
For instance, the ordered pair (2, 1) is a solution of this
system. To check this, substitute 2 for x and 1 for y in each
equation.
Check (2, 1) in Equation 1 and Equation 2:
2x + y = 5
2(2) + 1 ≟ 5
4+1=5
3x – 2y = 4
3(2) – 2(1) ≟ 4
6–2=4
Write Equation 1.
Substitute 2 for x and 1 for y.
Solution checks in Equation 1.
Write Equation 2.
Substitute 2 for x and 1 for y.
Solution checks in Equation 2.
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The Method of Substitution
You will study four ways to solve systems of equations,
beginning with the method of substitution.
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Example 1 – Solving a System of Equations by Substitution
Solve the system of equations.
x+y=4
Equation 1
x–y=2
Equation 2
Solution:
Begin by solving for y in Equation 1.
y=4–x
Solve for y in Equation 1.
Next, substitute this expression for y into Equation 2 and
solve the resulting single variable equation for x.
x–y=2
Write Equation 2.
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Example 1 – Solution
x – (4 – x) = 2
cont’d
Substitute 4 – x for y.
x–4+x=2
Distributive Property
2x = 6
Combine like terms.
x=3
Divide each side by 2.
Finally, solve for y by back-substituting x = 3 into the
equation y = 4 – x, to obtain
y=4–x
Write revised Equation 1.
y=4–3
Substitute 3 for x.
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Example 1 – Solution
y = 1.
cont’d
Solve for y.
The solution is the ordered pair (3, 1). Check this solution
as follows.
Check
Substitute (3, 1) into Equation 1:
x+y=4
3+1≟4
4=4
Write Equation 1.
Substitute for x and y.
Solution checks in Equation 1.
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Example 1 – Solution
cont’d
Substitute (3, 1) into Equation 2:
x–y=2
3–1≟2
2=2
Write Equation 2.
Substitute for x and y.
Solution checks in Equation 2.
Because (3, 1) satisfies both equations in the system, it is a
solution of the system of equations.
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The Method of Substitution
The term back-substitution implies that you work
backwards.
First you solve for one of the variables, and then you
substitute that value back into one of the equations in the
system to find the value of the other variable.
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Nonlinear Systems of Equations
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Nonlinear Systems of Equations
The equation in Example 1 is linear.
The method of substitution can also be used to solve
systems in which one or both of the equations are
nonlinear.
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Example 3 – Substitution: Two-Solution Case
Solve the system of equations.
3x2 + 4x – y = 7
Equation 1
2x – y = –1
Equation 2
Solution:
Begin by solving for y in Equation 2 to obtain y = 2x + 1.
Next, substitute this expression for y into Equation 1 and
solve for x.
3x2 + 4x – (2x + 1) = 7
3x2 + 2x – 1 = 7
Substitute 2x + 1 for y in Equation 1.
Simplify.
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Example 3 – Solution
3x2 + 2x – 8 = 0
(3x – 4)(x + 2) = 0
cont’d
Write in general form.
Factor.
Solve for x.
Back-substituting these values of x to solve for the
corresponding values of y produces the solutions
and
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Graphical Approach to Finding
Solutions
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Graphical Approach to Finding Solutions
A system of two equations in two unknowns can have
exactly one solution, more than one solution, or no solution.
By using a graphical method, you can gain insight about
the number of solutions and the location(s) of the
solution(s) of a system of equations by graphing each of
the equations in the same coordinate plane.
The solutions of the system correspond to the points of
intersection of the graphs.
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Graphical Approach to Finding Solutions
For instance, the two equations in Figure 7.1 graph as two
lines with a single point of intersection; the two equations in
Figure 7.2 graph as a parabola and a line with two points of
intersection; and the two equations in Figure 7.3 graph as a
parabola and a line with no points of intersection.
One intersection point
Two intersection points
Figure 7.1
Figure 7.2
No intersection points
Figure 7.3
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Example 5 – Solving a System of Equations Graphically
Solve the system of equations.
y = ln x
x+y=1
Equation 1
Equation 2
Solution:
There is only one point of
intersection of the graphs of the
two equations, and (1, 0) is the
solution point (see Figure 7.4).
Figure 7.4
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Example 5 – Solution
cont’d
Check this solution as follows.
Check (1, 0) in Equation 1:
y = ln x
Write Equation 1.
0 = ln 1
Substitute for x and y.
0=0
Solution checks in Equation 1.
Check (1, 0) in Equation 2:
x+y=1
1+0=1
1=1
Write Equation 2.
Substitute for x and y.
Solution checks in Equation 2.
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Graphical Approach to Finding Solutions
Example 5 shows the benefit of a graphical approach to
solving systems of equations in two variables.
Notice that by trying only the substitution method in
Example 5, you would obtain the equation x + ln x = 1.
It would be difficult to solve this equation for x using
standard algebraic techniques.
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Applications
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Applications
The total cost C of producing x units of a product typically
has two components—the initial cost and the cost per unit.
When enough units have been sold so that the total
revenue R equals the total cost C, the sales are said to
have reached the break-even point.
You will find that the break-even point corresponds to the
point of intersection of the cost and revenue curves.
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Example 6 – Break-Even Analysis
A shoe company invests $300,000 in equipment to produce
a new line of athletic footwear. Each pair of shoes costs $5
to produce and sells for $60. How many pairs of shoes
must the company sell to break even?
Solution:
The total cost of producing x units is
C = 5x + 300,000.
Equation 1
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Example 6 – Solution
cont’d
The revenue obtained by selling x units is
R = 60x.
Equation 2
Because the break-even point occurs when R = C, you
have C = 60x, and the system of equations to solve is
C = 5x + 300,000
C = 60x
.
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Example 6 – Solution
cont’d
Solve by substitution.
60x = 5x + 300,000
Substitute 60x for C in Equation 1.
55x = 300,000
Subtract 5x from each side.
x ≈ 5455
Divide each side by 55.
So, the company must sell about 5455 pairs of shoes to
break even.
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Applications
Another way to view the solution in Example 6 is to
consider the profit function
P = R – C.
The break-even point occurs when the profit is 0, which is
the same as saying that R = C.
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