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Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Systems of Linear
Equations and
Inequalities
15
Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
1. Decide whether a given ordered pair is a
solution of a system.
2. Solve linear systems by graphing.
3. Solve special systems by graphing.
4. Identify special systems without graphing.
Objectives
15.1 Solving Systems of Linear Equations by
Graphing
Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
A system of linear equations, often called a linear system,
consists of two or more linear equations with the same
variables.
Decide Whether a Given Ordered Pair is a Solution
2x + 3y = 4
3x – y = –5
or x + 3y = 1
–y = 4 – 2x
or x – y = 1
y = 3
A solution of a system of linear equations is an ordered pair
that makes both equations true at the same time. A solution of
an equation is said to satisfy the equation.
Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example Is (2,–1) a solution of the system 3x + y = 5
2x – 3y = 7 ?
Substitute 2 for x and –1 for y in each equation.
Decide Whether a Given Ordered Pair is a Solution
3(2) + (–1) = 5 ?
6 – 1 = 5 ?
5 = 5 True
2(2) – 3(–1) = 7 ?
4 + 3 = 7 ?
7 = 7 True
Since (2,–1) satisfies both equations, it is a solution of the
system.
Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example Is (2,–1) a solution of the system x + 5y = –3
4x + 2y = 1 ?
Substitute 2 for x and –1 for y in each equation.
Decide Whether a Given Ordered Pair is a Solution
2 + 5(–1) = – 3?
2 – 5 = –3?
–3 = –3 True
4(2) + 2(–1) = 1?
8 – 2 = 1?
6 = 1 False
(2,–1) is not a solution of this system because it does not
satisfy the second equation.
Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Solve the system of equations by graphing both equations on
the same axes.
Rewrite each equation in slope-intercept form to graph.
Solve Linear Systems by Graphing
–2x + ⅔y = –4 becomes y = 3x – 6 y-intercept (0, – 6); m = 3
5x – y = 8 becomes y = 5x – 8 y-intercept (0, – 8); m = 5
2
2 4
3
5 8
x y
x y
   
 
Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
Graph both lines on the same axes and identify where
they cross.
Solve Linear Systems by Graphing
y = 3x – 6 y = 5x – 8
-9
-8
-7
-6
-5
-3
-1
-4
-2
1
42-2-4 531-1-3-5
Because (1,–3) satis-
fies both equations,
the solution set of this
system is {(1,–3)}.
Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
CAUTION
We recommend using graph paper and a straightedge when
solving systems of equations graphically. It may not be
possible to determine from the graph the exact coordinates of
the point that represents the solution, particularly if those
coordinates are not integers. The graphing method does,
however, show geometrically how solutions are found and is
useful when approximate answers will suffice.
Solve Linear Systems by Graphing
Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Solve each system by graphing.
(a) 3x + y = 4
6x + 2y = 1
3x + y = 4 becomes y = –3x + 4;
y-intercept (0, 4); m = –3
Solve Special Systems by Graphing
Rewrite each equation in slope-intercept form to graph.
6x + 2y = 1 becomes y = –3x + ½
y-intercept (0, ½); m = –3
Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
The graphs of these
lines are parallel and
have no points in
common. For such a
system, there is no
solution.
y = –3x + 4
Solve Special Systems by Graphing
y = –3x + ½
-5
-3
-1
-4
-2
1
3
5
2
4
42-2-4 531-1-3-5
Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
Solve each system by graphing.
(b) ½x + y = 3
2x + 4y = 12
½x + y = 3 becomes y = –½x + 3;
y-intercept (0, 3); m = –½
Solve Special Systems by Graphing
Rewrite each equation in slope-intercept form to graph.
2x + 4y = 12 becomes y = –½x + 3;
y-intercept (0, 3); m = –½
Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
The graphs of these two
equations are the same
line. Thus, every point on
the line is a solution of the
system, and the solution
set contains an infinite
number of ordered pairs
that satisfy the equations.
y = – ½x + 3
Solve Special Systems by Graphing
y = – ½x + 3
-5
-3
-1
-4
-2
1
3
5
2
4
42-2-4 531-1-3-5
Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Three Cases for Solutions of Linear Systems with
Two Variables
1. The graphs intersect at exactly one point, which
gives the (single) ordered-pair solution of the
system. The system is consistent, and the
equations are independent.
Solve Special Systems by Graphing
Slide - 14Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Three Cases for Solutions of Linear Systems with
Two Variables (cont)
2. The graphs are parallel lines. So, there is no
solution and the solution set is . The system is
inconsistent and the equations are independent.
Solve Special Systems by Graphing
0
Slide - 15Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Three Cases for Solutions of Linear Systems with
Two Variables (cont)
3. The graphs are the same line. There is an infinite
number of solutions, and the solution set is written
in set-builder notation. The system is consistent
and the equations are dependent.
Solve Special Systems by Graphing
Slide - 16Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Describe the system without graphing. State the number of
solutions
(a) 3x + 2y = 6
–2y = 3x – 5
Identify Special Systems without Graphing
Rewrite each equation in slope-intercept form.
3 2 6
2 3 6
3
3
2
 
  
  
x y
y x
y x
2 3 5
3 5
2 2
  
  
y x
y x
Slide - 17Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
Both lines have slope but have different y-intercepts, (0,3)
and . Lines with the same slope are parallel, so these
equations have graphs that are parallel lines. Thus, the system
has no solution.
Identify Special Systems without Graphing
3
2

5
0,
2
 
 
 

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15.1 solving systems of equations by graphing

  • 1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 2 Systems of Linear Equations and Inequalities 15
  • 2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 1. Decide whether a given ordered pair is a solution of a system. 2. Solve linear systems by graphing. 3. Solve special systems by graphing. 4. Identify special systems without graphing. Objectives 15.1 Solving Systems of Linear Equations by Graphing
  • 3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G A system of linear equations, often called a linear system, consists of two or more linear equations with the same variables. Decide Whether a Given Ordered Pair is a Solution 2x + 3y = 4 3x – y = –5 or x + 3y = 1 –y = 4 – 2x or x – y = 1 y = 3 A solution of a system of linear equations is an ordered pair that makes both equations true at the same time. A solution of an equation is said to satisfy the equation.
  • 4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Is (2,–1) a solution of the system 3x + y = 5 2x – 3y = 7 ? Substitute 2 for x and –1 for y in each equation. Decide Whether a Given Ordered Pair is a Solution 3(2) + (–1) = 5 ? 6 – 1 = 5 ? 5 = 5 True 2(2) – 3(–1) = 7 ? 4 + 3 = 7 ? 7 = 7 True Since (2,–1) satisfies both equations, it is a solution of the system.
  • 5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Is (2,–1) a solution of the system x + 5y = –3 4x + 2y = 1 ? Substitute 2 for x and –1 for y in each equation. Decide Whether a Given Ordered Pair is a Solution 2 + 5(–1) = – 3? 2 – 5 = –3? –3 = –3 True 4(2) + 2(–1) = 1? 8 – 2 = 1? 6 = 1 False (2,–1) is not a solution of this system because it does not satisfy the second equation.
  • 6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Solve the system of equations by graphing both equations on the same axes. Rewrite each equation in slope-intercept form to graph. Solve Linear Systems by Graphing –2x + ⅔y = –4 becomes y = 3x – 6 y-intercept (0, – 6); m = 3 5x – y = 8 becomes y = 5x – 8 y-intercept (0, – 8); m = 5 2 2 4 3 5 8 x y x y      
  • 7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example (cont) Graph both lines on the same axes and identify where they cross. Solve Linear Systems by Graphing y = 3x – 6 y = 5x – 8 -9 -8 -7 -6 -5 -3 -1 -4 -2 1 42-2-4 531-1-3-5 Because (1,–3) satis- fies both equations, the solution set of this system is {(1,–3)}.
  • 8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G CAUTION We recommend using graph paper and a straightedge when solving systems of equations graphically. It may not be possible to determine from the graph the exact coordinates of the point that represents the solution, particularly if those coordinates are not integers. The graphing method does, however, show geometrically how solutions are found and is useful when approximate answers will suffice. Solve Linear Systems by Graphing
  • 9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Solve each system by graphing. (a) 3x + y = 4 6x + 2y = 1 3x + y = 4 becomes y = –3x + 4; y-intercept (0, 4); m = –3 Solve Special Systems by Graphing Rewrite each equation in slope-intercept form to graph. 6x + 2y = 1 becomes y = –3x + ½ y-intercept (0, ½); m = –3
  • 10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example (cont) The graphs of these lines are parallel and have no points in common. For such a system, there is no solution. y = –3x + 4 Solve Special Systems by Graphing y = –3x + ½ -5 -3 -1 -4 -2 1 3 5 2 4 42-2-4 531-1-3-5
  • 11. Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example (cont) Solve each system by graphing. (b) ½x + y = 3 2x + 4y = 12 ½x + y = 3 becomes y = –½x + 3; y-intercept (0, 3); m = –½ Solve Special Systems by Graphing Rewrite each equation in slope-intercept form to graph. 2x + 4y = 12 becomes y = –½x + 3; y-intercept (0, 3); m = –½
  • 12. Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example (cont) The graphs of these two equations are the same line. Thus, every point on the line is a solution of the system, and the solution set contains an infinite number of ordered pairs that satisfy the equations. y = – ½x + 3 Solve Special Systems by Graphing y = – ½x + 3 -5 -3 -1 -4 -2 1 3 5 2 4 42-2-4 531-1-3-5
  • 13. Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Three Cases for Solutions of Linear Systems with Two Variables 1. The graphs intersect at exactly one point, which gives the (single) ordered-pair solution of the system. The system is consistent, and the equations are independent. Solve Special Systems by Graphing
  • 14. Slide - 14Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Three Cases for Solutions of Linear Systems with Two Variables (cont) 2. The graphs are parallel lines. So, there is no solution and the solution set is . The system is inconsistent and the equations are independent. Solve Special Systems by Graphing 0
  • 15. Slide - 15Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Three Cases for Solutions of Linear Systems with Two Variables (cont) 3. The graphs are the same line. There is an infinite number of solutions, and the solution set is written in set-builder notation. The system is consistent and the equations are dependent. Solve Special Systems by Graphing
  • 16. Slide - 16Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example Describe the system without graphing. State the number of solutions (a) 3x + 2y = 6 –2y = 3x – 5 Identify Special Systems without Graphing Rewrite each equation in slope-intercept form. 3 2 6 2 3 6 3 3 2         x y y x y x 2 3 5 3 5 2 2       y x y x
  • 17. Slide - 17Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Example (cont) Both lines have slope but have different y-intercepts, (0,3) and . Lines with the same slope are parallel, so these equations have graphs that are parallel lines. Thus, the system has no solution. Identify Special Systems without Graphing 3 2  5 0, 2      