Unit 4.1

1. 7.1
Solving
Systems of
Two Equations
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 The Method of Substitution
 Solving Systems Graphically
 The Method of Elimination
 Applications
… and why
Many applications in business and science can be
modeled using systems of equations.
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3. Solution of a System
A solution of a system of two equations in two
variables is an ordered pair of real numbers that
is a solution of each equation.
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4. Example Using the Substitution
Method
Solve the system using the substitution method.
2x  y  10
6x  4y  1
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5. Example Using the Substitution
Method
Solve the system using the substitution method.
2x  y  10
6x  4y  1
Solve the first equation for y.
2x  y  10
y  2x 10
Substitute the expression for y into the second equation:
6x  4(2x 10)  1
x 
41
14
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y  2x 10

6. Example Using the Substitution
Method
6x  4(2x 10)  1
x 
41
14
y  2x 10
 2
41
14

 

 
10
 
29
7
The solution is the ordered pair
41
14
, 
29
7

 

 
.
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7. Example Solving a Nonlinear System
Algebraically
Solve the system algebraically.
y  x2  6x
y  8x
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8. Example Solving a Nonlinear System
Algebraically
y  x2  6x
y  8x
Substitute the values of y from the first equation into
the second equation:
8x  x2  6x
0  x2  2x
x  0, x  2.
If x  0, then y  0. If x  2, then y  16.
The system of equations has two solutions: (0,0) and (2,16).
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9. Example Using the Elimination
Method
Solve the system using the elimination method.
3x  2y  12
4x  3y  33
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10. Example Using the Elimination
Method
Solve the system using the elimination method.
3x  2y  12
4x  3y  33
Multiply the first equation by 3 and the second
equation by 2 to obtain:
9x  6y  36
8x  6y  66
Add the two equations to eliminate the variable y.
17x  102 so x  6
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11. Example Using the Elimination
Method
Solve the system using the elimination method.
3x  2y  12
4x  3y  33
Substitue x  6 into either of the two original equations:
3(6)  2y  12
2y  6
y  3
The solution of the original system is (6,  3).
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12. Example Finding No Solution
Solve the system:
3x  2y  5
6x  4y  10
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13. Example Finding No Solution
Solve the system:
3x  2y  5
6x  4y  10
Multiply the first equation by 2.
6x  4y  10
6x  4y  10
Add the equations:
0  20
The last equation is true for no values of x and y.
The equation has no solution.
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14. Example Finding Infinitely Many
Solutions
Solve the system.
3x  6y  10
9x 18y  30
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15. Example Finding Infinitely Many
Solutions
Solve the system.
3x  6y  10
9x 18y  30
Multiply the first equation by  3.
9x 18y  30
9x 18y  30
Add the two equations.
0  0
The last equation is true for all values of x and y.
The system has infinitely many solutions.
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16. Quick Review
1. Solve for y in terms of x. 2x  3y  6
Solve the equation algebraically.
2. x3  9x 3. x2  5x  6
4. Write the equation of the line that contains the point
(1,1) and is perpendicular to the line 2x  3y  6.
5. Write an equation equivalent to x  y  5 with
coefficient of x equal to  2.
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17. Quick Review
1. Solve for y in terms of x. 2x  3y  6 y  
2
3
x  2
Solve the equation algebraically.
2. x3  9x 0, 3 3. x2  5x  6  6,1
4. Write the equation of the line that contains the point
(1,1) and is perpendicular to the line 2x  3y  6.
y 1 
3
2
(x 1)
5. Write an equation equivalent to x  y  5 with
coefficient of x equal to  2.  2x  2y  10
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