3.1 Systems of Linear Equations

• Using graphs and tables to solve systems

• Using substitution and elimination to solve systems

• Using systems to model data

• Value, interests, and mixture problems

• Using linear inequalities in one variable to make predictions

Using Two Models to Make a Prediction

• When will the life expectancy of men and women be equal?– L = W(t) = 0.114t + 77.47– L = M(t) = 0.204t + 69.90

Years since 1980

Yea

rs o

f Li

fe

60

80

100

20 40 60 80 100 120

(84.11, 87.06)

Equal at approximately 87 years old in 2064.

System of Linear Equations in Two Variables (Linear System)

• Two or more linear equations containing two variables

y = 3x + 3

y = -x – 5

Solution of a System

• An ordered pair (a,b) is a solution of a linear system if it satisfies both equations.

• The solution sets of a system is the set of all solutions for that system.

• To solve a system is to find its solution set.

• The solution set can be found by finding the intersection of the graphs of the two equations.

• Graph both equations on the same coordinate plane– y = 3x + 3– y = -x – 5

• Verify– (-3) = 3(-2) + 3– -3 = -6 + 3– -3 = -3

– (-3) = -(-2) – 5 – -3 = 2 – 5 – -3 = -3

• Only one point satisfiesboth equations

• (-2,-3) is the solution set of the system

Find the Ordered Pairs that Satisfy Both Equations

Solutions for

y = 3x + 3

Solutions for

y = -x – 5

Solution for both (-2,-3)

Example• ¾x + ⅜y = ⅞• y = 3x – 5 • Solve first equation for y

– ¾x + ⅜y = ⅞– 8(¾x + ⅜y) = 8(⅞)– 24x + 24y = 56

4 8 8

– 6x -6x + 3y = 7 – 6x– 3y = -6x + 7

3 3 3

– y = -2x + 7/3

(1.45,-.6)

y = 3x – 5

-.6 = 3(1.45) – 5

-.6 = 4.35 – 5

-.6 ≈ -.65

y = -2x + 7/3

-.6 = -2(1.45) + 7/3

-.6 = -2.9 + 7/3

-.6 ≈ -.57

Inconsistent System

• A linear system whose solution set is empty– Example…Parallel lines never intersect

• no ordered pairs satisfy both systems

Dependent System

• A linear system that has an infinite number of solutions– Example….Two equations of the same line

• All solutions satisfy both lines

y = 2x – 2

-2x + y = -2• -2x +2x + y = -2 +2x• y = 2x – 2

One Solution System

• There is exactly one ordered pair that satisfies the linear system– Example…Two lines

that intersect in only

one point

Solving Systems with Tables

x 0 1 2 3 4

y = 4x – 6 -6 -2 2 6 10

y = -6x + 14 14 8 2 -4 -10

• Since (2,2) is a solution to both equations, it is a solution of the linear system.

• Isolate a variable on one side of either equation

• Substitute the expression for the variable into the other equation

• Solve the second equation

• Substitute the solution into one of the equations

3.2 Using Substitution

1. y = x – 1

2. 3x + 2y = 13• 3x + 2(x – 1) = 13• 3x + 2x – 2 = 13• 5x – 2 +2 = 13 +2• 5x = 15

5 5• x = 3

• y = 3 – 1 • y = 2

Solution set for the linear system is (3,2).

Example 1

Example 2

1. 2x – 6y = 4

2. 3x – 7y = 8• 2x – 6y +6y = 4 +6y• 2x = 6y + 4

2 2 2

• x = 3y + 2

• 3(3y + 2) – 7y = 8• 9y + 6 – 7y = 8• 2y + 6 -6 = 8 -6• 2y = 2

2 2

• y = 1• x = 3(1) + 2• x = 5

The solution set is (5, 1).

Using Elimination

• Adding left and right sides of equations

• If a=b and c=d, then a + c = b + d

– Substitute a for b and c for d• a + c = a + c• both sides are the same

Using Elimination

• Multiply both equations by a number so that the coefficients of one variable are equal in absolute value and opposite sign.

• Add the left and right sides of the equations to eliminate a variable.

• Solve the equation.

• Substitute the solution into one of the equations and solve.

Example 1

5x – 6y = 9

+ 2x + 6y = 12

7x + 0 = 21

7x = 21

7 7 x = 3

2(3) + 6y = 12

6 -6 + 6y = 12 -6

6y = 12

6 6

y = 2

Solution set for the system is (3,2)

Example 21. 3x + 7y = 29

2. 6x – 12y = 32

-2(3x + 7y) = -2(29)

-6x – 14y = -58

+ 6x – 12y = 32

0 – 26y = -26

-26 -26

y = 1

• 3x + 7(1) = 29• 3x + 7 -7 = 29 -7• 3x = 22

3 3

x = 22/3

Solution (22/3, 1)

Example 3

1. 2x + 5y = 14

2. 7x – 3y = 8

3(2x + 5y) = 3(14)

6x + 15y = 42

5(7x – 3y) = 5(8)

35x – 15y = 40

6x + 15y = 42

+ 35x – 15y = 40

41x + 0 = 82

41 41

x = 2

2(2) + 5y = 14

4 -4 + 5y = 14 -4

5y = 10

5 5

y = 2Solution (2,2)

Using Elimination with Fractions

2x – 5y -1 3 9 3

3x – 2y 715 3 5

9( ) = ( )9

15( ) = ( )15

-2(6x – 5y) = (-3)-2

-12x + 10y = 6+___________

-9x + 0 = 27

-9 -9x = -3

3x – 10y = 21

6x – 5y = -3

6(-3) – 5y = -3

-18 +18 – 5y = -3 +18

-5y = 15 -5 -5 y = -3

Inconsistent Systems

• y = 3x + 3• y = 3x – 2 • 3x + 3 = 3x – 2

3x -3x + 3 = 3x -3x – 2

3 ≠ - 2 False

• y = 3x + 3

-1(y) = -1(3x + 3)• -y = -3x – 3

+ y = 3x – 2

0 = 0 – 2

0 ≠ -2 False

•If the result of substitution or elimination of a linear system is a false statement, then the system is inconsistent.

Dependent Systems

1. y = 3x – 4

2. -9x + 3y = -12• -9x + 3(3x – 4) = -12• -9x + 9x -12 = -12 • -12 = -12• True

• y -3x = 3x -3x – 4 • -3(-3x + y) = (-4)-3• 9x + -3y = 12

+ -9x + 3y = -12

0 = 0

True

• If the result of applying substitution or elimination to a linear system is a true statement, then the system is dependent.

Solving systems in one variable

• To solve an equation A = B, in one variable, x, where A and B are expressions,

• Solve, graph or use a table of the system– y = A– y = B

• Where the x-coordinates of the solutions of the system are the solutions of the equation A = B

Systems in one variable

• y = 4x – 3 • y = -x + 2• 4x – 3 = -x + 2• 4x +x – 3 = -x +x + 2• 5x – 3 +3 = 2 +3• 5x = 5• 5 5• x = 1

• y = 4(1) – 3 • y = 1

(1,1)

Graphing to solve equations in one variable

• -2x + 6 = 5/4x – 3 – (3,1)

• -2x + 6 = -4– (6,-4)

(3,1)

(6,-4)

Using Tables to Solve Equations in One Variable

• -2x + 7 = 4x – 5

• Solution (2,3)

x y y

-2 11 -13

-1 9 -9

0 7 -5

1 5 -1

2 3 3

3 1 7

4 -1 11

3.3 Systems to Model Data

• Predict when the life expectance of men and women will be the same.

• L = W(t) = .114t + 77.47• L = M(t) = .204t + 69.90• .114t -.114t + 77.47 = .204t + 69.90 -.114t• 77.47 -69.90 = .09t + 69.90 -69.90• 7.57 = .09t• t ≈ 84.11

Solving to Make Predictions

• In 1950, there were 4 women nursing students at a private college and 84 men. If the number of women nursing students increases by 13 a year and the number of male nursing students by 6 a year. What year will the number of male and female students be the same?

• A = W(t) = 13t + 4• A = M(t) = 6t + 84

Using substitution

• 13t + 4 = 6t + 84

• 13t + 4 -4 = 6t + 84 -4

• 13t -6t = 6t -6t + 80

• 7t = 80

7 7

• t ≈ 11.43

• Equal in 1961~1962