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Systems of Linear Equations and Inequalities

CHAPTER

9.1 Solving Systems of Linear Equations Graphically9.2 Solving Systems of Linear Equations by Substitution;

Applications9.3 Solving Systems of Linear Equations by Elimination;

Applications9.4 Solving Systems of Linear Equations in Three Variables;

Applications9.5 Solving Systems of Linear Equations Using Matrices9.6 Solving Systems of Linear Equations Using Cramer’s

Rule9.7 Solving Systems of Linear Inequalities

Solving Systems of Linear Equations Graphically9.19.1

1. Determine whether an ordered pair is a solution for a system of equations.

2. Solve a system of linear equations graphically.3. Classify systems of linear equations in two unknowns.

System of equations: A group of two or more equations.

Solution for a system of equations: An ordered set of numbers that makes all equations in the system.

5 (Equation 1)

3 4 8 (Equation 2)

Checking a Solution to a System of EquationsTo verify or check a solution to a system of equations, 1. Replace each variable in each equation with its

corresponding value. 2. Verify that each equation is true.

Example 1Determine whether each ordered pair is a solution to the system of equations.

a. (3, 2) b) (3, 7)Solutiona. (3, 2) x + y = 7 y = 3x – 2

3 + 2 = 7 2 = 3(3) – 2 1 = 7 2 = 11 False False

7 (Equation 1)

3 2 (Equation 2)

Because (3, 2) does not satisfy both equations, it is not a solution for the system.

continued

b. (3, 4) x + y = 7 y = 3x – 2 3 + 4 = 7 4 = 3(3) – 2

7 = 7 4 = 7True False

Because (3, 4) does not satisfy both equations, it is not a solution to the system of equations.

A system of two linear equations in two variables can have one solution, no solution, or an infinite number of solutions.

The graphs intersect at a single point. There is one solution.

The equations have the same slope, the graphs are parallel. There is no solution.

The graphs are identical. There are an infinite number of solutions.

Solving Systems of Equations GraphicallyTo solve a system of linear equations graphically,1. Graph each equation.

a. If the lines intersect at a single point, then the coordinates of that point form the solution.b. If the lines are parallel, there is no solution.c. If the lines are identical, there are an infinite

number of solutions, which are the coordinates of

all the points on that line.2. Check your solution.

Example 2aSolve the system of equations graphically.

SolutionGraph each equation.The lines intersect at a singlepoint, (2, 4).We can check the point in eachequation to verify and we will leave that to you.

2 (Equation 1)

2 4 12 (Equation 2)

y = 2 – x

2x + 4y = 12(2, 4)

Example 2bSolve the system of equations graphically.

SolutionGraph each equation.The lines appear to be parallel, which we can verify by comparing the slopes.

(Equation 1)

(Equation 2)3 4

The slopes are the same, so the lines are parallel. The system has no solution

Example 2cSolve the system of equations graphically.

SolutionGraph each equation.The lines appear to be identical.

(Equation 1

(Equation

The equations are identical. All ordered pairs along the line are solutions.

Consistent system of equations: A system of equations that has at least one solution.

Inconsistent system of equations: A system of equations that has no solution.

Classifying Systems of EquationsTo classify a system of two linear equations in two unknowns, write the equations in slope-intercept form and compare the slopes and y-intercepts. Consistent system with independent equations: The system has a single solution at the point of intersection.

The graphs are different. They have different slopes.

Consistent system with dependent equations: The system has an infinite number of solutions.

The graphs are identical. They have the same slope and same y-intercept.

Inconsistent system: The system has no solution.

The graphs are parallel lines. They have the same slope, but different y-intercepts.

Example 3For each of the systems of equations in Example 2, determine whether the system is consistent with independent equations, consistent with dependent equations, or inconsistent. How many solutions does the system have?2a.

The graphs intersected at a single point.

The system is consistent.

The equations are independent (different graphs), and the system has one solution: (2, 4).

2 (Equation 1)

2 4 12 (Equation 2)

Example 3For each of the systems of equations in Example 2, determine whether the system is consistent with independent equations, consistent with dependent equations, or inconsistent. How many solutions does the system have?2b.

The graphs were parallel lines. The system is inconsistent and has no solutions.

(Equation 1)

(Equation 2)3 4

Example 3For each of the systems of equations in Example 2, determine whether the system is consistent with independent equations, consistent with dependent equations, or inconsistent. How many solutions does the system have?2c.

The graphs coincide. The system is consistent (it has a solution) with dependent equations (same graph) and has an infinite number of solutions.

(Equation 1

(Equation

Which set of points is a solution to the system?

a) (–1, 1)

b) (–1, –1)

c) (0, 2)

d) (–3, 7)

Which set of points is a solution to the system?

a) (–1, 1)

b) (–1, –1)

c) (0, 2)

d) (–3, 7)

Solving Systems of Linear Equations by Substitution; Applications9.29.2

1. Solve systems of linear equations using substitution.2. Solve applications involving two unknowns using a

system of equations.

Solving Systems of Two Equations Using SubstitutionTo find the solution of a system of two linear equations using the substitution method,1. Isolate one of the variables in one of the equations.2. In the other equation, substitute the expression you found in step 1 for that variable.3. Solve this new equation. (It will now have only one variable.)4. Using one of the equations containing both variables, substitute the value you found in step 3 for that variable and solve for the value of the other variable.5. Check the solution in the original equations.

Example 1Solve the system of equations using substitution.

Solution Step 1: Isolate a variable in one equation. The second equation is solved for x.

Step 2: Substitute x = 1 – y for x in the first equation.

4 5 8)

continued

Step 3: Solve for y. (1 )4 5 8yy

4 4 5 8y y

Step 4: Solve for x by substituting 4 for y in one of the original equations.

1x y 1 4x

3x The solution is (3, 4).

Solution Step 1: Isolate a variable in one equation. Use either equation. 2x + y = 7

y = 7 – 2x

Step 2: Substitute y = 7 – 2x for y in the second equation.

(7 2 ) 1x x

continuedStep 3: Solve for x.

Step 4: Solve for y by substituting 2 for x in one of the original equations.

2 7x y 2(2) 7y

4 7y The solution is (2, 3).

7 2( ) 1xx

7 2 1x x 3 7 1x

3 6x 2x

Inconsistent Systems of EquationsThe system has no solution because the graphs of the equations are parallel lines.You will get a false statement such as 3 = 4.

Consistent Systems with Dependent EquationsThe system has an infinite number of solutions because the graphs of the equations are the same line.You will get a true statement such as 8 = 8.

SolutionSubstitute y = 3x – 5 into the first equation.

3 ( 5 43 )

3 3 5 4x x 5 4 False statement. The

system is inconsistent and has no solution.

SolutionSubstitute y = 4 – 3x into the second equation.

6 2( )4 3 8

6 8 6 8x x 6 6 8 8x x

8 8 True statement. The number of solutions is infinite.

Solving Applications Using a System of EquationsTo solve a problem with two unknowns using a system of equations,1. Select a variable for each of the unknowns.2. Translate each relationship to an equation. 3. Solve the system.

Example 6

Terry is designing a table so that the length is twice the width. The perimeter is to be 216 inches. Find the length and width of the table.

Understand We are given two relationships and are to find the length and width.

Plan Select a variable for the length and another variable for the width, translate the relationships to a system of equations, and then solve the system.

continued

Execute Let l represent the length and w the width.Relationship 1: The length is twice the width.

Translation: l = 2wRelationship 2: The perimeter is 216 inches.

Translation: 2l + 2w = 216

System: 2

2 2 216

continuedSolve.

Now find the value of l. l = 2w = 2(36) = 72

Answer The length should be 72 inches and the width 36 inches.

2 2 216

2( ) 2 162 2w w 4 2 216w w

6 216w 36w

Substitute 2w for l.

Combine like terms.

Divide both sides by 6 to isolate w.

Example 8Anita and Ernesto are traveling north in separate cars on the same highway. Anita is traveling at 55 miles per hour and Ernesto is traveling at 70 miles per hour. Anita passes Exit 54 at 1:30 p.m. Ernesto passes the same exit at 1:45 p.m. At what time will Ernesto catch up with Anita?Understand To determine what time Ernesto will catch up with Anita, we need to calculate the amount of time it will take him to catch up to her. We can then add the amount to 1:45.

continuedPlan and Execute Let x = Anita’s travel time after passing the exit. Let y = Ernesto’s travel time after passing the exit.

Relationship 1: Ernesto passes the exit 15 minutes after Anita; Anita will have traveled 15 minutes longer.

Translation: x = y + ¼ (1/4 of an hour)

Relationship 2: When Ernesto catches up, they will have traveled the same distance.

Translation: 55x = 70y

Category Rate Time Distance

Anita 55 x 55x

Ernesto 70 y 70y

continuedOur system:

Use substitution:

455 70

Substitute to find x.1

40.92 0.25

155 70

5555 70

13.75 15 y

0.92 y

continued

Answer Ernesto will catch up to Anita in a little over 1 hour (1.17, which is 1 hour 10 minutes). The time will be1:45 p.m. + 1 hr 10 minutes = 2:55 p.m.

CheckVerify both given relationships.

55(1.17) 70(0.92)

64.35 64.4

41.17 0.92 0.25

1.17 1.17

Solve the system.

a) (10, 3)

b) (1, 4)

c) (10, 14)

d) (3, 6)

Solve the system.

a) (10, 3)

b) (1, 4)

c) (10, 14)

d) (3, 6)

Solve the system.

a) (2, 3)

b) (2, 6)

c) (2, 6)

d) (3, 2)

7 4 10

Solve the system.

a) (2, 3)

b) (2, 6)

c) (2, 6)

d) (3, 2)

7 4 10

Solving Systems of Linear Equations by Elimination; Applications9.39.3

1. Solve systems of linear equations using elimination.2. Solve applications using elimination.

Example 1Solve the system of equations using elimination.Solution We can add the equations.

4 0 16x

Notice that y is eliminated, so we can easily solve for the value of x.

4x 4 16x

Divide both sides by 4 to isolate x.

Now that we have the value of x, we can find y by substituting 4 for x in one of the original equations. x + y = 9

4 + y = 9y = 5

The solution is (4, 5).

Example 2Solve the system of equations.Solution Because no variables are eliminated if we

add, we rewrite one of the equations so that it has a term that is the additive inverse of one of the terms in the other equation.

Multiply the first equation by 4.

3 4 11

8x y 4 4 4 8x y

4 4 32x y 4 4 32

3 4 11

7 21x 3x

7 0 21x Solve for y.x + y = 83 + y = 8 y = 5

Example 3Solve the system of equations.Solution Choose a variable to eliminate, y, then

multiply both equations by numbers that make the y terms additive inverses.

Multiply the first equation by 2.Multiply the second equation by 5.

4 5 19

8 10 38

15 10 45

Multiply by 2.

Multiply by 5.

Add the equations to eliminate y.

7 0 7x 7 7x

continued

Substitute x = 1 into either of the original equations. 4x – 5y = 194(1) – 5y = 19 4 – 5y = 19 –5y = 15

The solution is (1, –3).

4 5 19

Example 4Solve the system of equations.

Solution To clear the decimals in Equation 1, multiply by 100.To clear the fractions in Equation 2, multiply by 5.

0.03 0.02 0.03

Multiply by 100.

Multiply by 5.

Multiply equation 2 by 1 then combine the equations.

continued

Substitute to find y. 3 2 3x y ( 2 313 ) y

3 2 3y 2 6y

3y The solution is (1, 3).

Solving Systems of Two Linear Equations Using EliminationTo solve a system of two linear equations using the elimination method,1. Write the equations in standard form (Ax + By = C).2. Use the multiplication principle to clear fractions or decimals (optional).3. If necessary, multiply one or both equations by a number (or numbers) so that they have a pair of terms that are additive inverses.4. Add the equations. The result should be an equation in terms of one variable.5. Solve the equation from step 4 for the value of that variable.6. Using an equation containing both variables, substitute the value you found in step 5 for the corresponding variable and solve for the value of the other variable.7. Check your solution in the original equations.

Inconsistent Systems and Dependent Equations

When both variables have been eliminated and the resulting equation is false, such as 0 = 5, there is no solution. The system is inconsistent.

When both variables have been eliminated and the resulting equation is true, such as 0 = 0, the equations are dependent. There are an infinite number of solutions.

Example 5aSolve the system of equations.

SolutionNotice that the left side of the equations are additive inverses. Adding the equations will eliminate both variables.

False statement. The system is inconsistent and has no solution.

Example 5bSolve the system of equations.

SolutionTo eliminate y multiply the first equation by 2.

True statement. The equations are dependent. There are an infinite number of solutions.

Multiply by 2. 6 2 8

Example 7Mia sells concessions at a movie theatre. In one hour, she sells 78 popcorns. The popcorn sizes are small, which sell for $4 each, and large, which sell for $6 each. If her sales totaled $420, then how many of each size popcorn did she sell?Understand The unknowns are the number of each size popcorn. One relationship involves the number of popcorns (78 total), and the other relationship involves the total sales in dollars ($420).

continuedPlan and Execute

Let x represent the number of small popcorns. Let y represent the number of large popcorns.

Relationship 1: The total number sold is 78.Translation: x + y = 78

Relationship 2: The total sales revenue is $420.Translation: 4x + 6y = 420

Category Price Number Revenue

Small $4 x 4x

Large $6 y 6y

Use elimination; choose x.

4 6 420

4 4 312

4 6 420

Multiply by 4.

2 108y54y

Substitute to find x. 78

continued

Answer Mia sold 54 large popcorns and 24 small popcorns.

24 54 78

x y 4 6 420

4(24) 6(54) 420

96 324 420

420 420

Example 8How many milliliters of a 20% HCl solution and 50% HCl solution must be mixed together to make 500 milliliters of 35% HCl solution?

Understand The unknowns are the volumes of 20% and 50% solution that are mixed.

One relationship involves the concentrations of each solution in the mixture and the other relationship involves the total volume of the final mixture (500 ml).

continuedPlan and Execute Let x and y represent the two amounts to be mixed.

Relationship 1: The total volume is 500 ml.Translation: x + y = 500

Relationship 2: The combined volumes of HCl in the two mixtures is to be 35% of the total mixture.

Translation: 0.20x + 0.50y = 0.35(500)

Solution Concentration Volume Amount of HCl

20% 0.20 x 0.20x

50% 0.50 y 0.50y

35% 0.35 x + y 0.35(500)

Use elimination; choose x.

0.20 0.50 175

0.20 0.20 100

0.20 0.50 175

Multiply by 0.20.

0.30 75y

250y Substitute to find x. 50

continued

Answer Mixing 250 ml of 20% solution with 250 ml of 50% solution gives 500 ml of 35% solution.

250 250 500

500 500

0.20 0.50 175

0.20(250) 0.50(250) 175

50 125 175

175 175

Solve the system.

a) (2, 3)

b) (7, 0)

c) (–2, 3)

d) (5, 5)

Solve the system.

a) (2, 3)

b) (7, 0)

c) (–2, 3)

d) (5, 5)

Solve the system.

a) (0, 10)

b) (1, 5)

c) (–2, 3)

d) no solution

5 5 50

Solve the system.

a) (0, 10)

b) (1, 5)

c) (–2, 3)

d) no solution

5 5 50

Solve the system.

a) (1, 1)

b) (1, 5)

c) (–1, 1)

d) no solution

5 6 11

Solve the system.

a) (1, 1)

b) (1, 5)

c) (–1, 1)

d) no solution

5 6 11

Solving Systems of Linear Equations in Three Variables; Applications9.49.4

1. Determine whether an ordered triple is a solution for a system of equations.

2. Understand the types of solution sets for systems of three equations.

3. Solve a system of three linear equations using the elimination method.

4. Solve application problems that translate to a system of three linear equations.

Example 1Determine whether (2, –1, 3) is a solution of the system 4,

2 2 3,

4 2 3.

Solution In all three equations, replace x with 2, y with –1, and z with 3.

x + y + z = 4

2 + (–1) + 3 = 4

2x – 2y – z = 3

2(2) – 2(–1) – 3 = 3

– 4x + y + 2z = –3

– 4(2) + (–1) + 2(3) = –3

–3 = –3

TRUEBecause (2, 1, 3) satisfies all three equations in the system, it is a solution for the system.

Types of Solution Sets A Single Solution: If the planes intersect at a single point, that ordered triple is the solution to the system.

Types of Solution Sets Infinite Number of Solutions: If the three planes intersect along a line, the system has an infinite number of solutions, which are the coordinates of any point along that line.

Infinite Number of Solutions:

If all three graphs are the same plane, the system has an infinite number of solutions, which are the coordinates of any point in the plane.

Types of Solution Sets No Solution: If all of the planes are parallel, the system has no solution.

No Solution: Pairs of planes also can intersect, as shown. However, because all three planes do not have a common intersection, the system has no solution.

Example 2a

Solution

Solve the system using elimination.

We select any two of the three equations and work to get one equation in two variables. Let’s add equations (1) and (2):

(4)2x + 3y = 8Adding to eliminate z

Next, we select a different pair of equations and eliminate the same variable. Let’s use (2) and (3) to again eliminate z.

(5) x – y + 3z = 8

Multiplying equation (2) by 3

4x + 5y = 14.

3x + 6y – 3z = 6

Now we solve the resulting system of equations (4) and (5). That will give us two of the numbers in the solution of the original system,

4x + 5y = 14

2x + 3y = 8(5)

We multiply both sides of equation (4) by –2 and then add to equation (5):

Substituting into either equation (4) or (5) we find that x = 1.

4x + 5y = 14–4x – 6y = –16,

–y = –2 y = 2

Now we have x = 1 and y = 2. To find the value for z, we use any of the three original equations and substitute to find the third number z.

Let’s use equation (1) and substitute our two numbers in it:

We have obtained the ordered triple (1, 2, 3). It should check in all three equations.

x + y + z = 6

1 + 2 + z = 6z = 3.

The solution is (1, 2, 3).

Example 2b

Solution

Solve the system using elimination.3 9 6 3

The equations are in standard form.

(4)3x + 2y = 4 Adding

Eliminate z from equations (2) and (3).

Eliminate z from equations (1) and (2).Multiplying equation (2) by 6

3 9 6 3

12 6 6 12

Adding15x + 15y = 15

Eliminate x from equations (4) and (5).

3x + 2y = 415x + 15y = 15

Multiplying top by 5

15x – 10y = 2015x + 15y = 15

Adding5y = 5y = 1

Using y = 1, find x from equation 4 by substituting.

3x + 2y = 43x + 2(1) = 4

continued

Substitute x = 2 and y = 1 to find z.x + y + z = 22 – 1 + z = 2

1 + z = 2 z = 1

The solution is the ordered triple (2, 1, 1).

3 9 6 3

Solving Systems of Three Linear Equations Using Elimination

To solve a system of three linear equations with three unknowns using elimination.

1. Write each equation in the form Ax + By+ Cz = D.

2. Eliminate one variable from one pair of equations using the elimination method.

3. If necessary, eliminate the same variable from another pair of equations.

continued

4. Steps 2 and 3 result in two equations with the same two variables. Solve these equations using the elimination method.

5. To find the third variable, substitute the values of the variables found in step 4 into any of the three original equations that contain the third variable.

6. Check the ordered triple in all three original equations.

Example 3

Solution

Solve the system using elimination.3 1

The equations are in standard form.

2 6 2 2

(4)5y 4z = 0Adding

5y + 4z = 2 (5)

Eliminate y from equations (4) and (5).

5y 4z = 05y + 4z = 2

All variables are eliminated and the resulting equation is false, which means that this system has no solution; it is inconsistent.

continued

Example 4At a movie theatre, Kara buys one popcorn, two drinks and 2 candy bars, all for $12. Rebecca buys two popcorns, three drinks, and one candy bar for $17. Leah buys one popcorn, one drink and three candy bars for $11. Find the individual cost of one popcorn, one drink and one candy bar.Understand We have three unknowns and three relationships, and we are to find the cost of each.Plan Select a variable for each unknown, translate the relationship to a system of three equations, and then solve the system.

continuedExecute: p = popcorn, d = drink, and c = candyRelationship 1: one popcorn, two drinks and two

candy bars, cost $12Translation: p + 2d + 2c = 12

Relationship 2: two popcorns, three drinks, and one candy bar cost $17

Translation: 2p + 3d + c = 17Relationship 3: one popcorn, one drink and three

candy bars cost $11Translation: p + d + 3c = 11

Choose to eliminate p: Start with equations 1 and 3.

Choose to eliminate p: Start with equations 1 and 2.

2 2 12 (Equation 1)

2 3 17 (Equation 2)

3 11 (Equation 3)

Multiply by 1

2 2 12 2 2 12

3 11 3 11

p d c p d c

��

1 (Equation 4)d c

Multiply by 22 2 12 2 4 4 24

2 3 17 2 3 17

p d c p d d

p d c p d c

��

3 7 (Equation 5)d c

continuedUse equations 4 and 5 to eliminate d.

1 (Equation 4)d c 3 7 (Equation 5)d c 4 6

Substitute for c in d – c = 11.5

Substitute for c and d in p + d + 3c = 11

p + 2.5 + 3(1.5) = 11

p + 7 = 11

The cost of one candy bar is $1.50.The cost of one drink is $2.50.The cost of one popcorn is $4.00.

Determine if (2, –5, 3) is a solution to the given system.

a) Yes

3 5 16

Determine if (2, –5, 3) is a solution to the given system.

a) Yes

3 5 16

Solve the system.

a) (–2, 2, –5)

b) (–5, 2, –2)

c) (–2, 5, 2)

d) no solution

3 2 14

Solve the system.

a) (–2, 2, –5)

b) (–5, 2, –2)

c) (–2, 5, 2)

d) no solution

3 2 14

Solving Systems of Linear Equations Using Matrices9.59.5

1. Write a system of equations as an augmented matrix.2. Solve a system of linear equations by transforming its

augmented matrix into row echelon form.

Matrix: A rectangular array of numbers.

The following are examples of matrices:

1 1 6 9 1/ 2 01 2

, 4 , 4 3 1 7 67 5

0 6 5 9 1 2

The individual numbers are called elements.

The rows of a matrix are horizontal, and the columns are vertical.

column 1 column 2 column 3

Augmented Matrix: A matrix made up of the coefficients and the constant terms of a system. The constant terms are separated from the coefficients by a dashed vertical line.

Let’s write this system as an augmented matrix:

Example 1 2 8

SolutionWrite the augmented matrix.

2 1 1 8

1 1 1 1

1 2 1 2

Write as an augmented matrix.

Row OperationsThe solution of a system is not affected by the following row operations in its augmented matrix. 1. Any two rows may be interchanged. 2. The elements of any row may be multiplied (or divided) by any nonzero real number.3. Any row may be replaced by a row resulting from

adding the elements of that row (or multiples of that row) to a multiple of the elements of any other row.

Row echelon form: An augmented matrix whose coefficient portion has 1’s on the diagonal from upper left to lower right and 0s below the 1’s.

1 1 1 1

0 1 1 2

0 0 1 1

Solution

Example 2

Solve the following linear system by transforming its augmented matrix into row echelon form.

We write the augmented matrix. 3 1 7

We perform row operations to transform the matrix into echelon form.

3R2 + R13 1 7

0 10 10

The resulting matrix represents the system:

R2 10 3 1 7

continued3 1 7

0 10 10

Since y = 1, we can solve for x using substitution.

3x + 1 = 7

3x + y = 7

3x = 6

Example 3

SolutionWrite the augmented matrix:

2 1 1 8

1 1 1 1 .

1 2 1 2

Use the row echelon method to solve

Our goal is to transform

into the form

2 1 1 8

1 1 1 1

1 2 1 2

a b c d

Interchange Row 1 and Row 21 1 1 1

2 1 1 8

1 2 1 2

1 1 1 1

0 3 3 6

1 2 1 2

–2R1 + R2

continued

1 1 1 1

0 1 1 2

0 1 2 1

1 1 1 1

0 1 1 2

0 0 3 3

–R2 + R3

(1/3)R2

R1 + R3

1 1 1 1

0 1 1 2

0 0 1 1

(1/3)R3

continued

z = –1.

Substitute z into y – z = 2 y –(1) = 2

y + 1 = 2 y = 1

continued

Substitute y and z into x – y + z = 1 x – 1 – 1 = 1

x – 2 = 1 x = 3

The solution is (3, 1, –1).

Replace R2 in with R1 + R2.

10 10 4

8 14 2

10 10 4

Replace R2 in with R1 + R2.

10 10 4

8 14 2

10 10 4

Solve by transforming the augmented matrix into row echelon form.

a) (2, 1, 3) b) (3, 1, 2)

c) (3, 2, 1) d) No solution

Solve by transforming the augmented matrix into row echelon form.

a) (2, 1, 3) b) (3, 1, 2)

c) (3, 2, 1) d) No solution

Solving Systems of Linear Equations Using Cramer’s Rule9.69.6

1. Evaluate determinants of 2 2 matrices.2. Evaluate determinants of 3 3 matrices.3. Solve systems of equations using Cramer’s Rule.

Square matrix: A matrix that has the same number of rows and columns.

Every square matrix has a determinant.

Determinant of a 2 2 Matrix

If then det(A) = 1 1

1 2 2 12 2

a b a ba b

Example 1aFind the determinant of the following matrix.

Solution

3 14 17

(3)(1) (7)( 2)

Example 1bFind the determinant of the following matrix.

Solution

12 15 3

( 4)( 3) (5)(3)

Example 1cFind the determinant of the following matrix.

Solution

(1)(8) (2)(4)

Minor of an element of a matrix: The determinant of the remaining matrix when the row and column in which the element is located are ignored.

Example 2Find the minor of 4 in

Solution To find the minor of 4, we ignore its row and column (shown in blue) and evaluate the determinant of the remaining matrix (shown in red).

2 6 1 .

( 2)(1) (4)( 1)

Evaluating the Determinant of a 3 3 Matrix

2 2 2 1 2 331 2

minorminor minor

of of of

A a b c a a aaa a

2 2 1 1 1 11 2 3

3 3 3 3 2 2

b c b c b ca a a

b c b c b c

Example 3Find the determinant of

Solution Use the rule for expanding by minors along the first column, we have the following:

2 6 1 .

6 1 2 1 2 63 ( 4) ( 5)

3 1 4 1 4 3

minor minor minor3 ( 4) ( 5)

of 3 of 4 of 5

3(6 3) 4( 2 4) 5(6 24)

9 8 90 107

Cramer’s Rule

The solution to the system of linear equations

isa x b y c

a x b y c

c b Dx

The solution to the system of linear equations

1 1 1 1

2 2 2 2

3 3 3 3

a x b y c z d

d b c Dx

a b c D

Da d cy

a b c D

a b d Dz

a b c D

Solution

Example 4 6 2.

Use Cramer’s Rule to solve

First, we find D, Dx, and Dy.6 1

6( 3) (2)(1)

2( 3) ( 2)(1)

6( 2) (2)(2)

20 5xD

20 5yD

The solution is (1/5, 4/5).

Solution

Example 5 2 8

Use Cramer’s Rule to solve

We need to find D, Dx, Dy, and Dz.

1 1 1 1 1 1(2) (1) ( 1)

2 1 2 1 1 1

(2)( 1 2) (1)(1 2) ( 1)(1 1)

continued

1 2 1yD

2 2 1xD

1 1 1 1 1 1(8) (1) ( 2)

2 1 2 1 1 1

(8)( 1 2) (1)(1 2) ( 2)(1 1)

24 3 0 27

1 1 8 1 8 1(2) (1) ( 1)

2 1 2 1 1 1

(2)(1 2) (1)(8 2) ( 1)(8 1)

6 6 9 9

continued

1 2 2zD

27 9 93, 1, 1.

9 9 9yx z

DD Dx y z

The solution is (3, 1, –1). The check is left to the viewer.

1 1 1 8 1 8(2) (1) ( 1)

2 2 2 2 1 1

(2)(2 2) (1)( 2 16) ( 1)(1 8)

0 18 9 9

Find the determinant.

Solve using Cramer’s Rule.

a) (1, 4)

b) (1, 3)

c) (2, 3)

d) No solution

2 6 22

Solve using Cramer’s Rule.

a) (1, 4)

b) (1, 3)

c) (2, 3)

d) No solution

2 6 22

Solving Systems of Linear Inequalities9.79.7

1. Graph the solution set of a system of linear inequalities.2. Solve applications involving a system of linear

inequalities.

Solving a System of Linear Inequalities in Two Variables

To solve a system of linear inequalities in two variables, graph all of the inequalities on the same grid. The solution set for the system contains all ordered pairs in the region where the inequalities’ solution sets overlap along with ordered pairs on the portion of any solid line that touches the region of overlap.

Example 1aGraph the solution set for the system of inequalities.

SolutionGraph the inequalities on thesame grid.Both lines will be dashed.

Solution

Example 1bGraph the solution set for the system of inequalities.

SolutionGraph the inequalities on thesame grid.

Solution

Example 2—Inconsistent SystemGraph the solution set for the system of inequalities.

SolutionGraph the inequalities on thesame grid.

The slopes are equal, so the lines are parallel. Since the shaded regions do not overlap there is no solution for the system.

Example 3Mr. Reynolds is landscaping his yard with some trees and bushes. He would like to purchase at least 3 plants. The trees cost $40 and the bushes cost $20. He cannot spend more than $300 for the plants. Write a system of inequalities that describes what Mr. Reynolds could purchase, then solve the system by graphing.

Understand We must translate to a system of inequalities, and then solve the system.

continuedPlan and ExecuteLet x represent the trees and y represent the bushes.

Relationship 1: Mr. Reynolds would like to purchase at least 3 plants.

x + y 3

Relationship 2: Mr. Reynolds cannot spend more than $300.

40x + 20y 300

continuedAnswerSince Mr. Reynolds cannot purchase negative plants, the solution set is confined to Quadrant 1.

300x y

Any ordered pair in the overlapping region is a solution. Assuming that only whole trees and bushes can be purchased, only whole numbers would be in the solution set. For example: (4, 2); (3, 4)

Graph.

Copyright © 2011 Pearson Education, Inc. Systems of Linear Equations and Inequalities CHAPTER...

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