Intermediate Algebra Chapter 4

• Systems

• of

• Linear Equations

Objective

• Determine if an ordered pair is a solution for a system of equations.

System of Equations

• Two or more equations considered simultaneously form a system of equations.

1 1 1

2 2 2

a x b y c

a x b y c

Checking 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.

Graphing Procedure

• 1. Graph both equations in the same coordinate system.

• 2. Determine the point of intersection of the two graphs.

• 3. This point represents the estimated solution of the system of equations.

Graphing observations

• Solution is an estimate• Lines appearing parallel have to

be checked algebraically.• Lines appearing to be the same

have to be checked algebraically.

Classifying Systems

• Meet in Point – Consistent – independent

• Parallel – Inconsistent – Independent

• Same – Consistent - Dependent

Def: Dependent Equations

•Equations with identical graphs

Independent Equations

•Equations with different graphs.

Algebraic Check• Same Line

1 1 1

2 2 2

a x b y c

a x b y c

1 1 1

2 2 2

a b c

a b c

Algebraic Check• Parallel Lines

1 1 1

2 2 2

a x b y c

a x b y c

1 1 1

2 2 2

a b c

a b c

Algebraic Check• Meet in a point {(x,y)}

1 1 1

2 2 2

a x b y c

a x b y c

1 1 1

2 2 2

a b c

a b c

Calculator Method for Systems

• Solve each equation for y

• Input each equation into Y=

• Graph

• Set Window

• Use CalIntersect

Calculator Problem

2 3

2 4

2, 1

y x

x y

Calculator Problem 2

3 4 8

33

4

x y

y x

Calculator Problem 3

6 2 4

3 2

x y

y x

, | 3 2x y y x

Objective

•Solve a System of Equations using the Substitution Method.

Substitution Method• 1. Solve one equation for one variable

• 2. In other equation, substitute the expression found in step 1 for that variable.

• 3. Solve this new equation (1 variable)

• 4. Substitute solution in either original equation

• 5. Check solution in original equation.

Althea Gibson – tennis player

•“No matter what accomplishments you make, someone helped you.”

Intermediate Algebra

•The

•Elimination

•Method

Notes on elimination method

• Sometimes called addition method

• Goal is to eliminate on of the variables in a system of equations by adding the two equations, with the result being a linear equation in one variable.

• 1. Write both equations in ax + by = c form

• 2. If necessary, multiply one or both of the equations by appropriate numbers so that the coefficients of one of the variables are opposites.

Procedure for addition method cont.

• 3. Add the equations to eliminate a variable.

• 4. Solve the resulting equation

• 5. Substitute that value in either of the original equations and solve for the other variable.

• 6. Check the solution.

Procedure for addition method cont.

• Solution could be ordered pair.• If a false statement results i.e. 1 =

0, then lines are parallel and solution set is empty set. (inconsistent)

• If a true statement results i.e. 0 = 0, then lines are same and solution set is the line itself. (dependent)

Practice Problem

• Answer {(2,4)}

6

2 5 16

x y

x y

Practice Problem Hint: eliminate x first

• Answer {(-7/2,-4)}

4 3 2

6 7 7

x y

x y

Practice Problem

2 1

2 3

x y

x y

Practice problem

3 4 5

9 12 15

x y

x y

Special Note on Addition Method

• Having solved for one variable, one can eliminate the other variable rather than substitute.

• Useful with fractions as answers.

Practice Problem – eliminate one variable and than the other

• Answer: {(8/3,1/3)}

3 3 15

4 4 44 5

33 3

x y

x y

Confucius

•“It is better to light one small candle than to curse the darkness.”

Intermediate Algebra 4.2

• Systems• Of

• Equations• In

• Three Variables

Objective

•To use algebraic methods to solve linear equations in three variables.

Def: linear equation in 3 variables

• is any equation that can be written in the standard form ax + by +cz =d where a,b,c,d are real numbers and a,b,c are not all zero.

Def: Solution of linear equation in three variables

• is an ordered triple (x,y,z) of numbers that satisfies the equation.

Procedure for 3 equations, 3 unknowns

• 1. Write each equation in the form ax +by +cz=d

• Check each equation is written correctly.

• Write so each term is in line with a corresponding term

• Number each equation

Procedure continued:

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

• 3. Eliminate the same variable from another pair of equations.

• Number these equations

Procedure continued

• 4. Use the two new equations to eliminate a variable and solve the system.

• 5. Obtain third variable by back substitution in one of original equations

Procedure continued

•Check the ordered triple in all three of the original equations.

Sample problem 3 equations

(1) 2

(2) 2 2 1

(3) 3 2 1

x y z

x y z

x y z

Answer to 3 eqs-3unknowns

•{(-2,3,1)}

Bertrand Russell – mathematician (1872-1970)

• “Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform.”

Cramer’s Rule

• Objective: Evaluate determinants of 2 x 2 matrices

• Objective: Solve systems of equations using Cramer’s Rule

Determinant

det[ ]a b

If A then Ac d

a b

ad bcc d

Cramer’s rule intuitive• Each denominator, D is the

determinant of a matrix containing only the coefficients in the system. To find D with respect to x, we replace the column of s-coefficients in the coefficient matrix with the constants form the system. To find D with respect to y, replace the column of y-coefficients in the coefficient matrix sit the constant terms.

Sample Problem: Evaluate:

• Answer = 16

3 2

2 4

Sample Cramer’s Rule problem

• Solve by Cramer’s Rule

2 3 5

3 9

x y

x y

Cramer’s Rule Answer

11

22

33x

y

D

D

D

222

11

333

11

x

y

Dx

DD

yD

Senecca

• “It is not because things are difficult that we do not dare, it is because we do not dare that they are difficult.”

Intermediate Algebra 5.5

• Applications

• Objective: Solve application problems using 2 x 2 and 3 x 3 systems.

Mixture Problems

• ****Use table or chart

• Include all units

• Look back to test reasonableness of answer.

Sample Problem

• How many milliliters of a 10% HCl solution and 30% HCl solution must be mixed together to make 200 milliliters of 15% HCl solution?

Mixture problem equations

200

0.10 0.30 30

x y

x y

Mixture problem answers

• 150 mill of 10% sol

• 50 mill of 30% sol

• Gives 200 mill of 15% sol

Distance Problems

• Include Chart and/or picture• Note distance, rate, and time in

chart• D = RT and T = D/R and R=D/T• Include units • Check reasonableness of answer.

Sample Problem

• To gain strength, a rowing crew practices in a stream with a fairly quick current. When rowing against the stream, the team takes 15 minutes to row 1 mile, whereas with the stream, they row the same mile in 6 minutes. Find the team’s speed in miles per hour in still water and how much the current changes its speed.

Distance problem equations

0.25( ) 1

0.1( ) 1

4

10

x y

x y

x y

x y

Answer

• Team row 7 miles per hour in still water

• Current changes speed by 3 miles per hour

Joe Paterno – college football coach

•“The will to win is important but the will to prepare is vital.”