Lecture notes, lecture 4.1 PDF

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4.1B Systems of Equations

April 19, 2011

MATH 1010 ~ Intermediate Algebra

Chapter 4: SYSTEMS OF EQUATIONS

Section 4.1: SYSTEMS OF EQUATIONS

Objectives: ✥Determine if ordered pairs are solutions of systems of equations. ✥Solve systems of equations graphically ✥Solve systems of equations by substitution. ✥Use systems of equations to model and solve real life problems.

x+y=3 x-y=-2

4.1B Systems of Equations

April 19, 2011

Vocabulary : system of equations

solution

point of intersection

consistent

inconsistent

dependent

4.1B Systems of Equations

Three methods to solve a system of equations: 1. Graphing 2. Substitution 3. Elimination

April 19, 2011

4.1B Systems of Equations

Solve each system by graphing

a) x - y = 3 2x + 3y = 7

b) 2x + y = 3 2y = -4x + 8

April 19, 2011

4.1B Systems of Equations

Solve by substitution a) y = 2x + 1 3x + 2y = 16

b) x + y = 3 2y = 2x + 6

c) 2x + 5y = 15 y = -2/5 x

April 19, 2011

4.1B Systems of Equations

a) x - y = 5 2x = 2y + 10

b) y = -3/2 x + 4 3x + 2y = 3

April 19, 2011

4.1B Systems of Equations

Set up a set of equations and solve these problems.

a) The sum of two numbers is 160. The larger number is three times the smaller number.

b) The perimeter of a rectangle is 90 meters. The length is 1½ times the width. Find the dimensions of the rectangle.

April 19, 2011

4.1B Systems of Equations

c) Ten pounds of a nut mixture sells for $6.95 per pound. The mixture is made from two kinds of nuts; peanuts at $5.65 per pound and cashews at $8.95 per pound. How many pounds of each will be used in the mixture?

April 19, 2011

4.1 Systems of Equations

April 19, 2011 MATH 1010 ~ Intermediate Algebra

Chapter 4: SYSTEMS OF EQUATIONS

Section 4.1: SYSTEMS OF EQUATIONS

Objectives: ✥Determine if ordered pairs are solutions of systems of equations. ✥Solve systems of equations graphically ✥Solve systems of equations by substitution. ✥Use systems of equations to model and solve real life problems.

x +y=3 x -y = - 2

Vocabulary : system of equations

solution

point of intersection

consistent

inconsistent

dependent

4.1 Systems of Equations

April 19, 2011

Three methods to solve a system of equations: 1. Graphing 2. Substitution 3. Elimination

Solve each system by graphing

a) x - y = 3 2x + 3y = 7

b) 2x + y = 3 2y = -4x + 8

4.1 Systems of Equations

April 19, 2011

Solve by substitution

a) y = 2x + 1 3x + 2y = 16

b) x + y = 3 2y = 2x + 6

c) 2x + 5y = 15 y = -2/5 x

a) x - y = 5 2x = 2y + 10

b) y = -3/2 x + 4 3x + 2y = 3

4.1 Systems of Equations

April 19, 2011

Set up a set of equations and solve these problems.

a) The sum of two numbers is 160. The larger number is three times the smaller number. Find the two numbers.

b) The perimeter of a rectangle is 90 meters. The length is 1½ times the width. Find the dimensions of the rectangle.

c) Ten pounds of a nut mixture sells for $6.95 per pound. The mixture is made from two kinds of nuts; peanuts at $5.65 per pound and cashews at $8.95 per pound. How many pounds of each will be used in the mixture?

MATH 1010 ~ Intermediate Algebra

Chapter 4: SYSTEMS OF EQUATIONS

Section 4.2: LINEAR SYSTEMS IN TWO VARIABLES

Objectives: ✣ Solve systems of equations by elimination. ✣ Use systems of equations to solve real life problems.

3 drinks + 4 doughnuts = $10.00 2 drinks + 2 doughnuts = $ 6.00 How much is 1 doughnut?

1

The method of elimination 1. Obtain coefficients for x (or y) that are opposites by multiplying all terms of one or both equations by suitable non-zero constants. 2. Add the equations to eliminate one variable and solve the resulting equation for the remaining variable. 3. Back-substitute the value obtained in step 2 in either of the original equations and solve for the other variable. 4. Check your solution in both of the original equations.

4x - 5y = 13 3x - y = 7

3x + 9y = 8 2x + 6y = 7

2

Solve these systems by elimination. a) -x + 2y = 9 x + 3y = 16

b) 3y = 2x + 21 ⅔x = 50 + y

c) 4x = 6 + 5y 8x = 12 + 10y

3

Solve these applications by an appropriate method.

a) An SUV costs $26,445 and an average of $0.18 per mile to maintain. A hybrid model of the SUV costs $31,910 and $0.13 to maintain. After how many miles will the cost of the SUV exceed the cost of the hybrid?

b) A total of $1790 was made by selling 200 adult tickets and 316 children's tickets to a charity event. The next night a total of $937.50 was made by selling 100 adult tickets and 175 children's tickets. Find the price of each type of ticket.

4

4.2 Linear Systems

April 19, 2011 MATH 1010 ~ Intermediate Algebra

Chapter 4: SYSTEMS OF EQUATIONS

Section 4.2: LINEAR SYSTEMS IN TWO VARIABLES

Objectives: ✣ Solve systems of equations by elimination. ✣ Use systems of equations to solve real life problems.

3 drinks + 4 doughnuts = $10.00 2 drinks + 2 doughnuts = $ 6.00 How much is 1 doughnut?

The method of elimination 1. Obtain coefficients for x (or y) that are opposites by multiplying all terms of one or both equations by suitable constants. 2. Add the equations to eliminate one variable and solve the resulting equation. 3. Back-substitute the value obtained in step 2 in either of the original equations and solve for the other variable. 4. Check your solution in both of the original equations.

4x - 5y = 13 3x - y = 7

3x + 9y = 8 2x + 6y = 7

4.2 Linear Systems

April 19, 2011

Solve these systems by elimination. a) -x + 2y = 9 x + 3y = 16

b) 3y = 2x + 21 ⅔x = 50 + y

c) 4x = 6 + 5y 8x = 12 + 10y

Solve these applications by an appropriate method.

a) An SUV costs $26,445 and an average of $0.18 per mile to maintain. A hybrid model of the SUV costs $31,910 and $0.13 to maintain. After how many miles will the cost of the SUV exceed the cost of the hybrid?

b) A total of $1790 was made by selling 200 adult tickets and 316 children's tickets to a charity event. The next night a total of $937.50 was made by selling 100 adult tickets and 175 children's tickets. Find the price of each type of ticket.

4.3B Linear Systems, 3 Variables

April 19, 2011

MATH 1010 ~ Intermediate Algebra

Chapter 4: SYSTEMS OF EQUATIONS

4.3: LINEAR SYSTEMS IN 3 VARIABLES Objectives: ✧Solve systems of equations in row-echelon form by back-substituting. ✧Solve systems of equations using Gaussian elimination ✧Solve application problems using Gaussian elimination.

3x - 2y + 4z = - 8 7y - 2z = 6 3z = 12

4.3B Linear Systems, 3 Variables

April 19, 2011

Row echelon form for a system of equations:

Three Elementary Row Operations:

x - 2y + 3z = 9 y + 2z = 5 z=3

1. Interchange two rows. 2. Multiply one row by a non-zero constant. 3. Add a multiple of one row to another row. Use these operations to get this system of equations in row echelon form.

x - 2y + 3z = 5 -x + y + 5z = 4 2x - 3z = 0

4.3B Linear Systems, 3 Variables

Possible solutions to a system of equations in three variables:

April 19, 2011

4.3B Linear Systems, 3 Variables

Solve this system. x - 2y + 2z = 9 -x + 3y = -4 2x - 5y + z = 10

April 19, 2011

4.3B Linear Systems, 3 Variables

Solve this system.

x - 3y + z = 1 2x -y - 2z = 2 x + 2y - 3z = -1

April 19, 2011

4.3B Linear Systems, 3 Variables

Solve this system. x + y - 3z = -1 y-z = 0 -x + 2y = 1

April 19, 2011

4.3B Linear Systems, 3 Variables

April 19, 2011

Write a set of equations to solve this problem.

The measure of one angle of a triangle is two-thirds the measure of a second angle. The measure of the second angle is 12º greater than the measure of the third angle. Find the measures of the three angles of the triangle.

4.3 Linear Systems, 3 Variables

April 19, 2011 MATH 1010 ~ Intermediate Algebra

Chapter 4: SYSTEMS OF EQUATIONS

4.3: LINEAR SYSTEMS IN 3 VARIABLES Objectives: ✧Solve systems of equations in row-echelon form by back-substituting. ✧Solve systems of equations using Gaussian elimination ✧Solve application problems using Gaussian elimination.

3x - 2y + 4z = - 8 7y - 2z = 6 3z = 12

Row echelon form for a system of equations:

Three Elementary Row Operations:

x - 2y + 3z = 9 y + 2z = 5 z=3

1. Interchange two rows. 2. Multiply one row by a non-zero constant. 3. Add a multiple of one row to another row. Use these operations to get this system of equations in row echelon form.

x - 2y + 3z = 5 -x + y + 5z = 4 2x - 3z = 0

4.3 Linear Systems, 3 Variables Possible solutions to a system of equations in three variables:

Solve this system. x - 2y + 2z = 9 -x + 3y = -4 2x - 5y + z = 10

April 19, 2011

4.3 Linear Systems, 3 Variables

April 19, 2011

Solve this system.

x - 3y + z = 1 2x -y - 2z = 2 x + 2y - 3z = -1

Solve this system. x + y - 3z = -1 y- z = 0 -x + 2y = 1

4.3 Linear Systems, 3 Variables

Write a set of equations to solve this problem.

The measure of one angle of a triangle is two-thirds the measure of a second angle. The measure of the second angle is 12º greater than the measure of the third angle. Find the measures of the three angles of the triangle.

April 19, 2011...