copyright © 2010 pearson education, inc. all rights reserved. 4.1 – slide 1


Systems of Linear Equations and Inequalities

Chapter 4


4.1

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.

4.1 Solving Systems of Linear Equations by Graphing


A system of linear equations, often called a linear system, consists of two or more linear equations with the same variables.

Deciding Whether a Given Ordered Pair is a Solution

2x + 3y = 43x – y = –5

or x + 3y = 1–y = 4 – 2x

or x – y = 1y = 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.



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.


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.


Example 2

Solve the system of equations by graphing both equations on the same axes.

Rewrite each equation in slope-intercept form to graph.

Solving 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


Example 2 (concluded)

Graph both lines on the same axes and identify where they cross.


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).



CAUTIONA difficulty with the graphing method of solution is that it may not be possible to determine from the graph the exact coordinates of the point that repre- sents the solution, particularly if these coordinates are not integers. For this reason, algebraic methods of solution are explained later in this chapter. The graphing method does, however, show geometrically how solutions are found and is useful when approximate answers will do.




Example 3

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

Solving Special Systems by Graphing


6x + 2y = 1 becomes y = –3x + ½

y-intercept (0, ½); m = –3



Example 3 (continued)

The graphs of these lines are parallel and have no points in common. For such a system, there is no solution.

y = –3x + 4


y = –3x + ½

-5

-3

-1

-4

-2

1

3

5

2

4

42-2-4 531-1-3-5



Example 3 (continued)

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



2x + 4y = 12 becomes y = –½x + 3; y-intercept (0, 3); m = –½



Example 3 (concluded)

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


y = – ½x + 3

-5

-3

-1

-4

-2

1

3

5

2

4

42-2-4 531-1-3-5



Three Cases for Solutions of Systems

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.




Three Cases for Solutions of Systems (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.



0


Three Cases for Solutions of Systems (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.



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