7.1
LINEAR AND NONLINEAR SYSTEMS OF EQUATIONS
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What You Should Learn
• 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.
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The Method of Substitution
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The Method of Substitution
2𝑥 + 𝑦 = 5
3𝑥 − 2𝑦 = 4
Equation 1
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, you can 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
Method
Type of System
1. Substitution
Linear or nonlinear, two variables
2. Graphical method
Linear or nonlinear, two variables
3. Elimination
Linear, two variables
4. Gaussian elimination
Linear, three or more variables
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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:
y=4–x
x–y=2
x – (4 – x) = 2
x–4+x=2
2x = 6
x=3
Solve for y in Equation 1.
Write Equation 2.
Substitute 4 – x for y.
Distributive Property
Combine like terms.
Divide each side by 2.
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Example 1 – Solution
cont’d
y=4–x
Write revised Equation 1.
y=4–3
y=1
Substitute 3 for x.
Check
Substitute (3, 1) into Equations 1 and 2:
x+y=4
x–y=2
3+1≟4
3–1≟2
4=4
2=2
The solution is the ordered pair (3, 1).
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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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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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Example
−𝑥 + 𝑦 = 4
𝑥2 + 𝑦 = 3
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Example
𝑥 2 + 𝑦 2 = 25
𝑥 − 8 2 + 𝑦 2 = 41
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Graphical Approach to Finding Solutions
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Graphical Approach to Finding Solutions
One intersection point
Two intersection points
No intersection points
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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:
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Example 5 – Solution
cont’d
You can 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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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 is sold for $60. How many pairs of shoes
must be sold before the business breaks 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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