solving systems of linear equations by graphing decide whether a given ordered pair is a solution of...

Solving Systems of Linear Equations by Graphing

Decide whether a given ordered pair is a solution of a system.

Solve linear systems by graphing.

Solve special systems by graphing.

Identify special systems without graphing.

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3

4

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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. Examples of systems include

2 3 4

3 5

x y

x y

3 4

4 2

x y

y x

1

3.

x y

y

Linear systems

In the system on the right, think of y = 3 as an equation in two variables by writing it as 0x + y = 3.

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Objective 1

Decide whether a given ordered pair is a solution of a system.

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Decide whether a given ordered pair is a solution of a system.A solution of a system of a linear equations is an ordered pair that makes both equations true at the same time. A solution is said to satisfy the equation.

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Solution:

Decide whether the ordered pair (4,−1) is a solution of each system.

5 6 14

2 5 3

x y

x y

3

3

x y

x y

5 4 6 1 14

2 4 5 1 3

4 1 3

4 1 3

20 6 14

8 5 3

14 14

3 3

4 1 3

4 1 3

3 3

3 3

Yes No

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EXAMPLE 1 Determining Whether an Ordered Pair is a Solution

Objective 2

Solve linear systems by graphing.

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Solve linear systems by graphing.

The set of all ordered pairs that are solutions of a system is its solution set.

Any intersection point would be on both lines and would therefore be a solution of both equations. Thus, the coordinates of any point at which the lines intersect give a solution of the system.

Because the two different straight lines can intersect at no more then one point, there can never be more than one solution set for such a system.

One way to find the solution set of a system of two linear equations is to graph both equations on the same axes. The graph of each line shows points whose coordinates satisfy the equation of that line.

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Solving a Linear System by Graphing

Step 1: Graph each equation of the system on the same coordinate axes.

Step 2: Find the coordinates of the point of intersection of the graphs if possible. This is the solution of the system.

Step 3: Check the solution in both of the original equations. Then write the solution set.

A difficulty with the graphing method is that 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 answer will do.

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Solve linear systems by graphing. (cont’d)

Solve the system by graphing.

Solution: {(3,2)}

5 3 9

2 7

x y

x y

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EXAMPLE 2 Solving a System by Graphing

Objective 3

Solve special systems by graphing.

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Solve special systems by graphing.

Sometimes the graphs of the two equations in a system either do not intersect at all or are the same line.

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When a system has an infinite number of solutions, either equation of the system could be used to write the solution set. It’s best to use the equation (in standard form) with coefficients that are integers having no common factor (except 1).

Solution:

3 4

6 2 12

x y

x y

Solve each system by graphing

, 2 5 8x y x y

2 5 8

4 10 16

x y

x y

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EXAMPLE 3 Solving Special Systems by Graphing

Three Cases for Solutions of Systems1. 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. See below left.

2. The graphs are parallel lines, so there is no solution and the solution set is Ø. The system is inconsistent and the equations are independent. See below middle.

3. The graphs are the same line. There is an infinite number of solutions, and the solution set is written in set-builder notation as {(x,y)|_________}, where one of the equations is written after the | symbol. The system is consistent and equations are dependent. See below right.

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Solve special systems by graphing. (cont’d)

Objective 4

Identify special systems without graphing.

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Identify special systems without graphing.

Example 3 showed that the graphs of an inconsistent system are parallel lines and the graphs of a system of dependent equations are the same line. We can recognize these special kinds of systems without graphing by using slopes.

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Solution:

The equations represent parallel lines. The system has no solution.

2 3 5

3 2 7

x y

y x

2 5

3 32 7

3 3

y x

y x

1 2

3 31 2

3 3

y x

y x

Describe each system without graphing. State the number of solutions.

6 3

2 11

x y

x y

3 2

2 6 4

x y

x y

6 3

2 11

y x

y x

The equations represent the same line. The system has an infinite number of solutions.

The equations represent lines that are neither parallel nor the same line. The system has exactly one solution.

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EXAMPLE 4 Identifying the Three Cases by Using Slopes