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3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

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Page 1: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

3.1 Solve Linear Systems by Graphing

Note: Definitely need graph paper for your notes today

Page 2: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

System of Two Linear Equations

• Also called a linear system

• Simply two linear equations

• Solution of the system: an ordered pair that satisfies both equations

Page 3: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

EXAMPLE 1 Solve a system graphically

Graph the linear system and estimate the solution. Then check the solution algebraically.

4x + y = 8

2x – 3y = 18

Equation 1

Equation 2

SOLUTION

Begin by graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (3, – 4). You can check this algebraically as follows.

Page 4: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

EXAMPLE 1 Solve a system graphically

Equation 1 Equation 2

4x + y = 8

4(3) + (– 4) 8=?

=?12 – 4 8

8 = 8

2x – 3y = 18

=?2(3) – 3( – 4) 18

=?6 + 12 18

18 = 18

The solution is (3, – 4).

Page 5: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

SOLUTION

GUIDED PRACTICE for Example 1

Graph the linear system and estimate the solution. Then check the solution algebraically.1. 3x + 2y = – 4

x + 3y = 13x + 2y = – 4x + 3y = 1

Equation 1Equation 2

Begin by graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (–2, 1). You can check this algebraically as follows.

Page 6: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

GUIDED PRACTICE for Example 1

Equation 1 Equation 23x + 2y = –4

=?–6 + 2 –4

x + 3y = 1

–2 + 3 1=?

1 = 1

The solution is (–2, 1).

=?3(–2) + 2(1) –4

–4 = –4

(–2 ) + 3( 1) 1=?

Page 7: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

SOLUTION

GUIDED PRACTICE for Example 1

Graph the linear system and estimate the solution. Then check the solution algebraically.

Equation 1Equation 2

2. 4x – 5y = – 102x – 7y = 4

2x – 7y = 44x – 5y = – 10

Begin by graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (–5, –2). You can check this algebraically as follows.

Page 8: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

GUIDED PRACTICE for Example 1

Equation 1 Equation 2 2x –7y = 4

–10 + 14 4=?

4 = 4

The solution is (–5, –2).

–10 = –10

4x – 5y = –10=?4(–5) – 5(–2 ) –10 2(–5) – 7(–2 ) 4=?

=?–20 + 10 –10

Page 9: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

SOLUTION

GUIDED PRACTICE for Example 1

Graph the linear system and estimate the solution. Then check the solution algebraically.

Equation 1Equation 2

3. 8x – y = 83x + 2y = – 16

8x – y = 83x + 2y = – 16

Begin by graphing both equations, as shown at the right. From the graph, the lines appear to intersect at (0, –8). You can check this algebraically as follows.

Page 10: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

GUIDED PRACTICE for Example 1

Equation 1 Equation 2

=? 0 + 8 8 0 – 16 –16=?

–16 = –16

The solution is (0, –8).

8(0) – (–8 ) 8=?

8 = 8

8x – y = 8 3x + 2y = –163(0) + 2(–8) –16=?

Page 11: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

How to do it with a graphing calculator

Page 12: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

Classifying Systems page 154

• Consistent: at least one solution

• Inconsistent: no solution (lines never intersect)

Page 13: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

Types of Consistent Systems

• Independent- exactly one solution (graphs have one intersection point)

• Dependent- infinitely many solutions (two graphs are the same line)

Page 14: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

Keep in Mind…

• If it’s inconsistent- parallel lines, neither independent or dependent

• If it’s consistent- either one intersection point or two equations that represent the same line- classify as either independent or dependent

Page 15: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

EXAMPLE 2 Solve a system with many solutions

Solve the system. Then classify the system as consistent and independent,consistent and dependent, or inconsistent.

4x – 3y = 8

8x – 6y = 16

Equation 1

Equation 2

SOLUTION

The graphs of the equations are the same line. So, each point on the line is a solution, and the system has infinitely many solutions. Therefore, the system is consistent and dependent.

Page 16: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

If it is an inconsistent system (no solution)

• The lines are parallel– The lines have the same _______.

Page 17: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

EXAMPLE 3 Solve a system with no solution

Solve the system. Then classify the system as consistent and independent,consistent and dependent, or inconsistent.

2x + y = 4

2x + y = 1

Equation 1

Equation 2

SOLUTION

The graphs of the equations are two parallel lines.Because the two lines have no point of intersection, the system has no solution. Therefore, the system is inconsistent.

Page 18: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

Guided Practice 4 – 6

Page 19: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

EXAMPLE 4 Standardized Test Practice

SOLUTION

Equation 1 (Option A)

y = 1 x + 30

Page 20: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

Equation 2 (Option B)

EXAMPLE 4 Standardized Test Practice

y = x2.5

To solve the system, graph the equations y = x + 30 and y = 2.5x, as shown at the right.

Page 21: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

EXAMPLE 4 Standardized Test Practice

Notice that you need to graph the equations only in the first quadrant because only nonnegative values of x and y make sense in this situation.

The lines appear to intersect at about the point (20, 50). You can check this algebraically as follows.

Equation 1 checks.

Equation 2 checks.

50 = 20 + 30

50 = 2.5(20)

ANSWERThe total costs are equal after 20 rides.

The correct answer is B.

Page 22: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

Extra Word Problem Example

• A soccer league offers two options for membership plans. Option A incluces an initial fee of $40 and costs $5 for each game played. Option B costs $10 for each game played. After how many games will the total cost of the two options be the same?

Page 23: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

Review Questions

• How do you solve a linear system by graphing?

• How can you tell just by looking at the system that the same graph is shown twice?

Page 24: 3.1 Solve Linear Systems by Graphing Note: Definitely need graph paper for your notes today

Homework

• 1, 2, 6-11, 15, 16, 20 – 25, 28-31, 35 – 39