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7.1 
Solving 
Systems of 
Two Equations 
Copyright © 2011 Pearson, Inc.
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. 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 2
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 - 3
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 - 4
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 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 5 
y  2x 10
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 
 
  
 
  
. 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 6
Example Solving a Nonlinear System 
Algebraically 
Solve the system algebraically. 
y  x2  6x 
y  8x 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 7
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). 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 8
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 - 9
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 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 10
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). 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 11
Example Finding No Solution 
Solve the system: 
3x  2y  5 
6x  4y  10 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 12
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. 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 13
Example Finding Infinitely Many 
Solutions 
Solve the system. 
3x  6y  10 
9x 18y  30 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 14
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. 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 15
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 - 16
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 
Copyright © 2011 Pearson, Inc. Slide 7.1 - 17

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Unit 7.1

  • 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. Copyright © 2011 Pearson, Inc. Slide 7.1 - 2
  • 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 - 3
  • 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 - 4
  • 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 Copyright © 2011 Pearson, Inc. Slide 7.1 - 5 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       . Copyright © 2011 Pearson, Inc. Slide 7.1 - 6
  • 7. Example Solving a Nonlinear System Algebraically Solve the system algebraically. y  x2  6x y  8x Copyright © 2011 Pearson, Inc. Slide 7.1 - 7
  • 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). Copyright © 2011 Pearson, Inc. Slide 7.1 - 8
  • 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 - 9
  • 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 Copyright © 2011 Pearson, Inc. Slide 7.1 - 10
  • 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). Copyright © 2011 Pearson, Inc. Slide 7.1 - 11
  • 12. Example Finding No Solution Solve the system: 3x  2y  5 6x  4y  10 Copyright © 2011 Pearson, Inc. Slide 7.1 - 12
  • 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. Copyright © 2011 Pearson, Inc. Slide 7.1 - 13
  • 14. Example Finding Infinitely Many Solutions Solve the system. 3x  6y  10 9x 18y  30 Copyright © 2011 Pearson, Inc. Slide 7.1 - 14
  • 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. Copyright © 2011 Pearson, Inc. Slide 7.1 - 15
  • 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 - 16
  • 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 Copyright © 2011 Pearson, Inc. Slide 7.1 - 17