Chapter 10: Systems of Equations and Inequalities

10.1 Systems of Linear Equations;Substitution

or Elimination (Stack/Add)

1. Definitions

An equation is linear if it can be written as:

System of linear equations. Collection of 2 or more equations containing one or more variables.

Solution to a system of equations. The values of the variables which make all the equations true.

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Definition

Definition

Definition

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22

yx

yxExample

1. More Definitions

Consistent – System with at least one solution

Two types of Consistent Solutions:• Dependent- System with infinitely many

solutions• Independent– System with only one solution

Inconsistent – System with no solutions

Definition

Definition

2. Verify a solutionVerify that (4,-1) is a solution to:

Is (-4,3) a solution ?

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22

yx

yx

3. Methods for Solving System of Linear Equations

1) Substitution

2) Elimination (Stack/Add)

3) Graphing (For system of 2 variables)

4) Section 10.2: Matrices

What does the graph of this equation look like?

22 yx

123 zyx

4. Method of SubstitutionGoal: Convert to equation of one variable.

Verify Solution when finished!

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yx

yx

5. Method of Elimination

Goal: Add 2 equations together to eliminate a variableUse: When a variable cannot be easily isolated

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yx

yx#1

5. Method of Elimination (Stack/Add)

Verify Solution!

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yx

yx#2

1) Equations should be of form: Ax + By = C and variables lined up2) Multiply by nonzero number so a variable cancels when adding3) Add the equations 4) Solve the new equation5) Back-substitute

6. Inconsistent System

What is the graph of this system?

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623

yx

yx#3 There is no solution when the

result is a false statement with no variables involved

Examples : 0 = 2-1 = 5

7. Dependent system

The solution of a Dependent System is a set: {(x,y) | }

10515

23

yx

xy#4 Infinitely many solutions

if your solution results in a statement that is always true.

Examples:2 = 20 = 0

-3/4 = -3/4

8. Applications

Solving an application problem:

Step 1: Define the variables

Step 2: Write the system in words (describe verbally) Step 3: Plug in variables for the words.

Step 4: Solve the system.

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9. System of 3 Linear EquationsGOAL: Reduce the system to 2 equations in the same 2

variables and solve for the 2 variables.

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822

932

zyx

zyx

zyx

10. Example of Dependent 3x3Solve:

What does solution look like on graph?Write Solution as {(x,y) | }

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zyx

zyx

zyx

10. System with missing termSolve:

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zyx

zyx

zx (1) is already in 2 variables, x and z

add equations (2) and (3) to cancel y.

11. Curve Fitting

cbxaxy 2

(-1,-2), (1,-4) and (2,4) are points that lie on the graph of a quadratic.

Determine the coefficients a, b, and c

Polya’s 4 Principles for solving word problems

1. Understand the problem: Read carefully (annotate, highlight)• What are you asked to find or show ? (variables)• Can you restate the problem in your own words?• Can you think of a picture/diagram/table to help you understand

the problem?• Do you understand all the words used in stating the problem?

Do you need to ask a question to get the answer? ( Hint: “I don’t understand” is not a question!)

2. Devise a plan• Make a list, look for a pattern, work backward, be creative

3. Carry out the plan• Persistence and patience pay off.

4. Review/extend: Reflect on what worked and what didn’t to predict the strategy to use in future problems.

Traffic Control:

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