Download Chapter 12: Systems of Linear Equations and Inequalities

12
Systems of Linear
Equations and Inequalities
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 1
12.1 Solving Systems of Linear Equations by
Graphing
Objectives
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 2
Decide Whether a Given Ordered Pair is a Solution
A system of linear equations, often called a linear system,
consists of two or more linear equations with the same
variables.
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 3
Decide Whether a Given Ordered Pair is a Solution
Example 1 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.
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 4
Decide Whether a Given Ordered Pair is a Solution
Example 1 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.
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 5
Solve Linear Systems by Graphing
Example 2
Solve the system of equations by graphing both equations on
the same axes.
2
2x 
y  4
3
5x  y  8
Rewrite each equation in slope-intercept form to graph.
–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
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 6
Solve Linear Systems by Graphing
Example 2 (concluded)
y = 3x – 6
y = 5x – 8
Graph both lines on the same axes and identify where
they cross.
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1
2
3
4
5
Because (1,–3) satisfies both equations,
the solution set of this
system is {(1,–3)}.
-7
-8
-9
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 7
Solve Linear Systems by Graphing
CAUTION
With the graphing method, it may not be possible to
determine from the graph the exact coordinates of the
point that represents the solution, particularly if these
coordinates are not integers. The graphing method
does, however, show geometrically how solutions are
found and is useful when approximate answers will
suffice.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 8
Solve Special Systems by Graphing
Example 3
Solve each system by graphing.
(a) 3x + y = 4
6x + 2y = 1
Rewrite each equation in slope-intercept form to graph.
3x + y = 4 becomes y = –3x + 4;
y-intercept (0, 4); m = –3
6x + 2y = 1 becomes y = –3x + ½
y-intercept (0, ½); m = –3
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 9
Solve Special Systems by Graphing
Example 3 (continued)
y = –3x + 4
y = –3x + ½
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
1
2
3
4
5
The graphs of these
lines are parallel and
have no points in
common. For such a
system, there is no
solution.
-4
-5
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 10
Solve Special Systems by Graphing
Example 3 (continued)
Solve each system by graphing.
(b) ½x + y = 3
2x + 4y = 12
Rewrite each equation in slope-intercept form to graph.
½x + y = 3 becomes y = –½x + 3;
y-intercept (0, 3); m = –½
2x + 4y = 12 becomes y = –½x + 3;
y-intercept (0, 3); m = –½
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 11
Solve Special Systems by Graphing
Example 3 (concluded)
y = – ½x + 3
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
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
-2
-3
-4
-5
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 12
Solve Special Systems by Graphing
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 13
Solve Special Systems by Graphing
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 0. The system is
inconsistent and the equations are independent.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 14
Solve Special Systems by Graphing
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.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 15
Identify Special Systems without Graphing
Example 4
Describe the system without graphing. State the number of
solutions
(a) 3x + 2y = 6
–2y = 3x – 5
Rewrite each equation in slope-intercept form.
3x  2 y  6
2 y  3x  6
3
y   x3
2
2 y  3 x  5
3
5
y  x
2
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 16
Identify Special Systems without Graphing
Example 4 (continued)
3
Both lines have slope  but have different y-intercepts, (0,3)
2
5

and  0,  . Lines with the same slope are parallel, so these
 2
equations have graphs that are parallel lines. Thus, the system
has no solution.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Slide 17