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The discussed topic in the ppt is: Solving Linear Equations

1.
Chapter 2

Equations and

Inequalities

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-1

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Equations and

Inequalities

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-1

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2.
Chapter Sections

2.1 – Solving Linear Equations

2.2 – Problem Solving and Using Formulas

2.3 – Applications of Algebra

2.4 – Additional Application Problems

2.5 – Solving Linear Inequalities

2.6 – Solving Equations and Inequalities

Containing Absolute Values

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-2

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2.1 – Solving Linear Equations

2.2 – Problem Solving and Using Formulas

2.3 – Applications of Algebra

2.4 – Additional Application Problems

2.5 – Solving Linear Inequalities

2.6 – Solving Equations and Inequalities

Containing Absolute Values

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-2

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3.
§ 2.1

Solving Linear

Equations

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-3

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Solving Linear

Equations

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-3

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4.
Properties of Equality

Properties of Equality

For all real numbers a, b, ,and c:

1. a = a Reflexive property

2. If a = b, then b = a Symmetric property

3. If a = b, and b = c, then a = c Transitive property

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-4

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Properties of Equality

For all real numbers a, b, ,and c:

1. a = a Reflexive property

2. If a = b, then b = a Symmetric property

3. If a = b, and b = c, then a = c Transitive property

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-4

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5.
Combine Like Terms

Like terms are terms that have the same

variables with the same exponents.

Like Terms Unlike Terms

1

-3x, 8x, - 3 x 20x, x2, x3

6w2, -12w2, w2 6xy, 2xyz, w2

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Like terms are terms that have the same

variables with the same exponents.

Like Terms Unlike Terms

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-3x, 8x, - 3 x 20x, x2, x3

6w2, -12w2, w2 6xy, 2xyz, w2

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-5

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6.
Combining Like Terms

1. Determine which terms are like terms.

2. Add or subtract the coefficients of the like

terms.

3. Multiply the number found in step 2 by the

common variable(s).

Example: 5a + 7a = 12a

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1. Determine which terms are like terms.

2. Add or subtract the coefficients of the like

terms.

3. Multiply the number found in step 2 by the

common variable(s).

Example: 5a + 7a = 12a

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-6

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7.
Distributive Property

For any real numbers a, b, and c,

a(b + c) = ab + bc

Example: 3(x + 5) = 3x + 15

(This is not equal to 18x! These are

not like terms.)

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For any real numbers a, b, and c,

a(b + c) = ab + bc

Example: 3(x + 5) = 3x + 15

(This is not equal to 18x! These are

not like terms.)

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-7

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8.
Simplifying an Expression

1. Use the distributive property to remove

any parentheses.

2. Combine like terms.

Simplify 3(x + y) + 2y

= 3x + 3y + 2y (Distributive Property)

= 3x + 5y (Combine Like Terms)

(Remember that 3x + 5y cannot be combined because

they are not like terms.)

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1. Use the distributive property to remove

any parentheses.

2. Combine like terms.

Simplify 3(x + y) + 2y

= 3x + 3y + 2y (Distributive Property)

= 3x + 5y (Combine Like Terms)

(Remember that 3x + 5y cannot be combined because

they are not like terms.)

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-8

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9.
Solve Linear Equations

A linear equation in one variable is an

equation that can be written in the

form ax + b = c where a, b, and c are real

numbers and a 0.

The solution to an equation is the

number that when substituted for the

variable makes the equation a true

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A linear equation in one variable is an

equation that can be written in the

form ax + b = c where a, b, and c are real

numbers and a 0.

The solution to an equation is the

number that when substituted for the

variable makes the equation a true

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-9

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10.
Addition Property of Equality

If a = b, then a + c = b + c for any

real numbers a, b, and c.

Example:

Solve the equation x – 4 = -10.

x – 4 = -10

x – 4 + 4 = -10 + 4 (Add 4 to both sides.)

x = -6

Check: (-6) – 4 = -10

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If a = b, then a + c = b + c for any

real numbers a, b, and c.

Example:

Solve the equation x – 4 = -10.

x – 4 = -10

x – 4 + 4 = -10 + 4 (Add 4 to both sides.)

x = -6

Check: (-6) – 4 = -10

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-10

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11.
Multiplication Property of Equality

If a = b, then a · c = b · c for any real

numbers a, b, and c.

Example: Solve the equation 12y = 15.

1 1 1

12 ·12y

1

= 15 · 12 (Multiply both sides by 12 )

5

1 12 y 15 1

12 12

1

y 5 4

4

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If a = b, then a · c = b · c for any real

numbers a, b, and c.

Example: Solve the equation 12y = 15.

1 1 1

12 ·12y

1

= 15 · 12 (Multiply both sides by 12 )

5

1 12 y 15 1

12 12

1

y 5 4

4

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-11

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12.
Multiplication Property of Equality

Example: Solve the equation 1 x 3 .

4 4

( 4)( 1 x) 3 ( 4) (Multiply both sides by -4)

4 4

( 4)( 1 x) 3 ( 4) (Simplify)

4 4

x = -3

1 (-3) 3

Check: 4

4

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Example: Solve the equation 1 x 3 .

4 4

( 4)( 1 x) 3 ( 4) (Multiply both sides by -4)

4 4

( 4)( 1 x) 3 ( 4) (Simplify)

4 4

x = -3

1 (-3) 3

Check: 4

4

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-12

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13.
Solve Linear Equations

1. Clear fractions. If the equation contains fractions, eliminate the fractions by

multiplying both sides of the equation by the least common denominator.

2. Simplify each side separately. Simplify each side of the equation as much as

possible. Use the distributive property to clear parentheses and combine like

terms as needed.

3. Isolate the variable term on one side. Use the addition property to get all

terms with the variable on one side of the equation and all constant terms on

the other side. It may be necessary to use the addition property a number of

times to accomplish this.

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1. Clear fractions. If the equation contains fractions, eliminate the fractions by

multiplying both sides of the equation by the least common denominator.

2. Simplify each side separately. Simplify each side of the equation as much as

possible. Use the distributive property to clear parentheses and combine like

terms as needed.

3. Isolate the variable term on one side. Use the addition property to get all

terms with the variable on one side of the equation and all constant terms on

the other side. It may be necessary to use the addition property a number of

times to accomplish this.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-13

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14.
Solve Linear Equations

4. Solve for the variable. Use the multiplication

property to get the variable (with a coefficient

of 1) on one side.

5. Check. Check by substituting the value

obtained in step 4 back into the original

equation.

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4. Solve for the variable. Use the multiplication

property to get the variable (with a coefficient

of 1) on one side.

5. Check. Check by substituting the value

obtained in step 4 back into the original

equation.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-14

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15.
Solving Equations

Example: Solve the equation 2x + 9 = 14.

2 x 9 14

2 x 9 9 14 9

2 x 5

2x 5

2 2

5

x

2 Don’t forget to check!

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-15

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Example: Solve the equation 2x + 9 = 14.

2 x 9 14

2 x 9 9 14 9

2 x 5

2x 5

2 2

5

x

2 Don’t forget to check!

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-15

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16.
Solve Equations Containing Fractions

2a

Example: Solve the equation 5 9

3

The least common denominator is 3.

2a

5 9 15 15 2a 27 15

3

2a 42

2a

3 5 3( 9) 2a 42

3

2 2

2a

3(5) 3 27 a 21

3

15 2a 27

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2a

Example: Solve the equation 5 9

3

The least common denominator is 3.

2a

5 9 15 15 2a 27 15

3

2a 42

2a

3 5 3( 9) 2a 42

3

2 2

2a

3(5) 3 27 a 21

3

15 2a 27

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-16

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17.
Identify Conditional Equations, Contradictions,

and Identities

Conditional Equations: Equations that true for only specific

values of the variable.

Contradictions: Equations that are never true and have no solution.

Identities: Equations that are always true and have an infinite

number of solutions.

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and Identities

Conditional Equations: Equations that true for only specific

values of the variable.

Contradictions: Equations that are never true and have no solution.

Identities: Equations that are always true and have an infinite

number of solutions.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-17

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18.
Solving Equations with Decimals

Example: Determine whether the equation 5(a - 3) – 3(a –

6) = 2(a + 1) + 1 is a conditional equation, a contradiction,

or an identity.

5(a 3) 3(a 6) 2(a 1) 1

5a 15 3a 18 2a 2 1

2a 3 2a 3

Since we obtain the same expression on both sides of the

equation, it is an identity.

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Example: Determine whether the equation 5(a - 3) – 3(a –

6) = 2(a + 1) + 1 is a conditional equation, a contradiction,

or an identity.

5(a 3) 3(a 6) 2(a 1) 1

5a 15 3a 18 2a 2 1

2a 3 2a 3

Since we obtain the same expression on both sides of the

equation, it is an identity.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 2-18

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