copyright © 2010 pearson education, inc. all rights reserved sec 2.1 - 1


Linear Equations and Applications

Chapter 2


2.1

Linear Equations in One Variable

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.1 - 4

2.1 Linear Equations in One Variable

Objectives

1. Decide whether a number is a solution of a linear

equation.

2. Solve linear equations using the addition and

multiplication properties of equality.

3. Solve linear equations using the distributive

property.

4. Solve linear equations with fractions or decimals.

5. Identify conditional equations, contradictions,

and identities.


2.1 Using Linear Equations of One Variable

Algebraic Expressions vs. Equations

In the previous chapter, we looked at algebraic expressions:

– 9y + 5, 10k, and

2 57

a

b c-Equations are statements that two algebraic expressions are equal:

3x – 13 = 29, 2 + y = – 11, and 3m = 4m – 2

An equation always contains an equals sign, but an expression does not.




Linear Equation in One Variable

A linear equation is also called a first-degree equation since the greatest power on the variable is one.

5x + 10 = 13

A linear equation in one variable can be written in the form Ax + B =

Cwhere A, B, and C are real numbers, with A = 0./




Determine whether the following equations are linear or nonlinear.

8x + 3 = –99x3 – 8 = 15

x7 = –

12

4 16 x

Yes, x is raised to the first power.

No, x is not raised to the first power.





Deciding Whether a Number is a Solution

If a variable can be replaced by a real number that makes the equation a true statement, then that number is a solution of the equation, x – 10 = 3.

x – 10 = 3

13

13 – 10 = 3

x – 10 = 3

8

8 – 10 = 3

(true) (false)

13 is a solution 8 is not a solution



Finding the Solution Set of an Equation

An equation is solved by finding its solution set – the set of all solutions.

The solution set of

x – 10 = 3is {13}.

Equivalent equations are equations that have the same solution set. These are equivalent equations since they all have solution set {–3}.

3x + 5 = –4

3x = –9 x = –3



Solving Linear Equations

An equation is like a balance scale, comparing the weights of two quantities.

Expression-1 Expression-2

We apply properties to produce a series of simpler equivalent equations to determine the solution set.

Variable Solution

=

=


C


Addition Property of Equality

The same number may be added to both sides of an equation without changing the solution set.

A =

=C+

A = B

+A B

B


C


Multiplication Property of Equality

Each side of an equation may be multiplied by the same nonzero number without changing the solution set.

A =

= C

A = B

A B

B



Addition and Multiplication Properties of Equality

For all real numbers A, B, and C, the equation

A = B and A + C = B + C

are equivalent.

Addition Property of Equality

For all real numbers A, B, and for C = 0, the equation

A = B and A C = B C

are equivalent.

Multiplication Property of Equality/



Addition and Multiplication Properties of Equality

Because subtraction and division are defined in terms of addition and multiplication,

we can extend the addition and multiplication properties of equality as follows:

The same number may be subtracted from each side of an equation, and each side of an equation may be divided by the same nonzero number, without changing the solution set.



Solving Linear Equations in One Variable

Step 1 Clear fractions. Eliminate any fractions by multiplying each side by the least common denominator.

Step 2 Simplify each side separately. Use the distributive property to clear parentheses and combine like terms as needed.

Step 3 Isolate the variable terms on one side. Use the addition property to get all terms with variables on one side of the equation and all numbers on the other.

Step 4 Isolate the variable. Use the multiplication property to get an equation with just the variable (with coefficient of 1) on one side.

Step 5 Check. Substitute the proposed solution into the original equation.




Solve 3x + 2 = 10.

3x + 2 = 10

3x + 2 – 2 = 10 – 2 3x = 8

Subtract 2.

Combine like terms.

Divide by 3.

Proposed solution.

3 8

3 3

x

8

3x




3x + 2 = 10

3 • + 2 = 10

38 Check by substituting the proposed

solution back into the original equation.

8 + 2 = 10Since the value of each side is 10, the proposed solution is correct.

The solution set is8

3

.




Solve 2x – 5 = 5x – 2.

2x – 5 = 5x – 2

2x – 5 – 5x = 5x – 2 – 5x

–3x – 5 = –2 Subtract 5x.

Combine like terms.

Add 5.

Divide by –3.

–3x – 5 + 5= –2 + 5

–3x = 3 Combine like terms.

x = –1 Proposed solution.

3 3

3 3

x




2x – 5 = 5x – 2

Check by substituting the proposed solution back into the original equation.

–2 – 5 = –5 – 2

Since the value of each side is –7 , the proposed solution is correct.

The solution set is {–1}.

2(–1) – 5 = 5(–1) – 2

–7 = –7




Solve 5(2x + 3) = 3 – 2(3x – 5). 5(2x + 3) = 3 – 2(3x – 5)

10x + 15 = 3 – 6x + 10

10x + 15 – 15 = 3 – 6x + 10 – 15

10x = – 6x – 2

10x + 6x = –6x – 2 + 6x

16x = –2

Distributive Prop.

Add –15.

Collect like terms.

Add 6x.

Collect like terms.




Divide by 16.

1

8x Proposed solution.

16x = –2

16 2

16 16

x




Check proposed solution:

5 2 3 3 2 3 5 22 435 3 2

8 8

110 865 2 3 3 2 3 5 3

8

110 110 Checks

8 8

The solution set se

8

2 3 110 24 865 3 3 2 5

8 8 8 8

1 1

8 8

t is

8

2 24 3 405 3 2

8 8 8 8

x x

1.

8



Solving Linear Equations with Fractions

Solve 2 1 1 3

2 3 4

x x.

Clear fractions.

Distributive property

2 1 1 312 12

2 3 4

6 2 1 4 3 3

1

.

Distributive property.

A

2 6 4 3 9

12 dd 3 .6 4 33 39

x x

x x

x x

xx x xx



Solving Linear Equations with Fractions

Collect like terms.

Add 6.

Coll

12 6 4 3 9 3

9

ect like terms

Divide by 9.

Proposed solu

6 5

9 6 5

9 1

tion

1

9 11

11

6 6

9

9.

9

x x x x

x

x

x

x

x

continued



Solving Linear Equations with Decimals

Solve ( )1.5 2 2.8x x+ = + .

Multiply by 10.


Add 10 .

Collect like terms.

Add 30.

1.5 2 2.8

15 2 28 10

15 30 28 10

15 30 28 10

5 30 28

10

5 30

1

830 32

0

0

x x

x x

x x

x

x

x

xxxx



Solving Linear Equations with Decimals

Collect like terms.

Divide by 5.

Proposed solution.

30

2The

5 30 28

5 2

5 2

2

solution set is

5

30

5 5

.5

x

x

x

x

continued



Conditional, Contradiction, and Identity Equations

Linear equations can have exactly one solution, no solution, or an infinite number of solutions.

Type of Linear Equation

Number of Solutions Indication When Solving

Conditional One Final results is x = a number.

Identity Infinite; solution set {all real numbers}

Final line is true, such as 5 = 5.

Contradiction None; solution set is Final line is false, such as –3 = 11..




A contradiction has no solutions.

Adding 7.

Collecting like terms

Sol 7 2.

7 2

7 2

.

Add .

Col

5

5

0 5 lecting like term .

7

s

ve

7

x x

x x

x x

x x

x xxxx

Since 0 = –5 is never true, and this equation is equivalent to x + 7 = x + 2, the solution set is empty.




An identity has an infinite number of solutions.


Adding 2.

Collecting like terms.

Adding 2 .

Collecting like

2Solv

te

e 2 2 1 .

2 2 2 1

2 2 2 2

2 2 2 2

2

2

2

2 2

rms.

2

0 0

2 2

x x

x x

x x

x x

x x

x xxx x

Since 0 = 0 is always true, and this equation is equivalent to 2x + 2 = 2(x + 1), the solution set is all real numbers.

copyright © 2010 pearson education, inc. all rights reserved sec 2.1 - 1

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