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Unit 4.1
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Copyright © 2011 Pearson, Inc.
7.1Solving
Systems of Two Equations
Copyright © 2011 Pearson, Inc. Slide 7.1 - 2
What you’ll learn about
The Method of Substitution Solving Systems Graphically The Method of Elimination Applications
… and whyMany applications in business and science can be modeled using systems of equations.
Copyright © 2011 Pearson, Inc. Slide 7.1 - 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.
Copyright © 2011 Pearson, Inc. Slide 7.1 - 4
Example Using the Substitution Method
Solve the system using the substitution method.
2x y 10
6x 4y 1
Copyright © 2011 Pearson, Inc. Slide 7.1 - 5
Example Using the Substitution Method
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
14y 2x 10
241
14
10
29
7
Solve the system using the substitution method.
2x y 10
6x 4y 1
Copyright © 2011 Pearson, Inc. Slide 7.1 - 6
Example Using the Substitution Method
6x 4(2x 10) 1
x 41
14y 2x 10
241
14
10
29
7
The solution is the ordered pair
41
14,
29
7
.
Copyright © 2011 Pearson, Inc. Slide 7.1 - 7
Example Solving a Nonlinear System Algebraically
Solve the system algebraically.
y x2 6x
y 8x
Copyright © 2011 Pearson, Inc. Slide 7.1 - 8
Example Solving a Nonlinear System Algebraically
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).
y x2 6x
y 8x
Copyright © 2011 Pearson, Inc. Slide 7.1 - 9
Example Using the Elimination Method
Solve the system using the elimination method.
3x 2y 12
4x 3y 33
Copyright © 2011 Pearson, Inc. Slide 7.1 - 10
Example Using the Elimination Method
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
Solve the system using the elimination method.
3x 2y 12
4x 3y 33
Copyright © 2011 Pearson, Inc. Slide 7.1 - 11
Example Using the Elimination Method
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).
Solve the system using the elimination method.
3x 2y 12
4x 3y 33
Copyright © 2011 Pearson, Inc. Slide 7.1 - 12
Example Finding No Solution
Solve the system:
3x 2y 5
6x 4y 10
Copyright © 2011 Pearson, Inc. Slide 7.1 - 13
Example Finding No Solution
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.
Solve the system:
3x 2y 5
6x 4y 10
Copyright © 2011 Pearson, Inc. Slide 7.1 - 14
Example Finding Infinitely Many Solutions
Solve the system.
3x 6y 10
9x 18y 30
Copyright © 2011 Pearson, Inc. Slide 7.1 - 15
Example Finding Infinitely Many Solutions
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.
Solve the system.
3x 6y 10
9x 18y 30
Copyright © 2011 Pearson, Inc. Slide 7.1 - 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.
Copyright © 2011 Pearson, Inc. Slide 7.1 - 17
Quick Review
1. Solve for y in terms of x. 2x 3y 6 y 2
3x 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 13
2(x 1)
5. Write an equation equivalent to x y 5 with
coefficient of x equal to 2. 2x 2y 10
