2.3 More on Solving Linear Equations

Objective 1

Learn and use the four steps for solving a linear equation.

Slide 2.3-3

We solve more complicated equations using the following four-step method.

Solving a Linear EquationStep 1: Simplify each side separately. Clear parentheses, fractions, and

decimals, using the distributive property as needed, and combine all like terms.

Step 3: Isolate the variable. Use the multiplication property if necessary to get the equation in the form x = a number.

(Other letters may be used for variables.)

Step 2: Isolate the variable term on one side. Use the addition property if necessary so that the variable term is on one side of the equation and a number is on the other.

Step 4: Check. Substitute the proposed solution into the original equation to see if a true statement results.

Slide 2.3-4

Learn and use the four steps for solving a linear equation.

CLASSROOM EXAMPLE 1

Solve.

Solution:

5 15

5 5

p

3p

Check:

5 4 19p

45 1 44 9p

5 15p

( 3)5 4 19 15 4 19

19 19

The solution set of the equation is {−3}.

Slide 2.3-5

Applying Both Properties of Equality to Solve an Equation

CLASSROOM EXAMPLE 2

Solve.

Solution:

55 10 5 5x

10

1 0

10

0 1x

1x

Check:

5 8 2 5x x

25 8 22 5xx xx 5 10 5x

(1)8 )5 2 5(1 3 3

The solution set of the equation is {1}.

Slide 2.3-6

Applying Both Properties of Equality to Solve an Equation

Remember that the variable can be isolated on either side of the equation. There are often several equally correct ways to solve an equation.

CLASSROOM EXAMPLE 3

Solve.

Solution:

1414 1 142 6x

2

2 2

2

x

1x

Check:

11 3 1 5 16x x

11 3 3 5 16x x

51 3 1 54 5 6x xx x

11 3 1 5 16x x

11 3 1 51 161

11 3 0 5 16 11 11

The solution set of the equation is {−1}.

Slide 2.3-7

Using Four Steps to Solve an Equation

CLASSROOM EXAMPLE 4

Solve.

Solution:

73 77 9x

3 6

3 3

1x

16

3x

Check:

4 ( 7) 9x x

7) 914 (x x

4 7 9x x

4 ( 7) 9x x

4 716

39

16

3

64 16 21 27

3 3 3 3 64 37 27

3 3 3

The solution set of the equation is16

.3

27 27

3 3

Slide 2.3-8

Using the Four Steps to Solve an Equation

Be very careful with signs when solving an equation like the one in the previous example. When clearing parentheses in the expression remember that the − sign acts like a factor of −1 and affects the sign of every term within the parentheses.

Slide 2.3-9

Learn and use the four steps for solving a linear equation. (cont’d)

CLASSROOM EXAMPLE 5

Solve.

Solution:

44 422 4z

22 2

2

2

2 0z

0z

Check:

2 3 2 6 4 1 8z z

2 6 18 4 4 8z z 44 418 4 4zz z z 2 3 2 6 1 80 04

2 3(2 0) 4 0 8 2 6 0 0 8 2 12 14

The solution set of the equation is {0}.

2 3 2 6 4 1 8z z

14 14

Slide 2.3-10

Using the Four Steps to Solve an Equation

Objective 2

Solve equations with fractions or decimals as coefficients.

Slide 2.3-11

We clear an equation of fractions by multiplying each side by the least common denominator (LCD) of all the fractions in the equation. It is a good idea to do this to avoid messy computations.

When clearing an equation of fractions, be sure to multiply every term on each side of the equation by the LCD.

Slide 2.3-12

Solve equations with fractions or decimals as coefficients.

CLASSROOM EXAMPLE 6

Solve.

Solution:

5 99 92x 14 2

2 2

x

7x

Check:1 5 3 1

3 12 4 2x x

44 5 6 49xx x x

33 33

12 12

The solution set of the equation is {−7}.

1 5 3 1

3 1212 12 12

21

42x x

1 5 3 1

3 12 27 7

4

7 5 3 7

3 12 4 2

28 5 9 42

12 12 12 12

1 5 3 1

3 12 4 2x x

Slide 2.3-13

Solving an Equation with Fractions as Coefficients

CLASSROOM EXAMPLE 7

Solve.

5x

Solution:

3( 3) 8( 1) 24x x

3 9 8 8 24x x

5 1 24x

1 23 1 ( 2)

4 312 12 12x x

The solution set of the equation is {5}.

1 23 112

41 2

32x x

1 23 1 2

4 3x x

Slide 2.3-14

Solving an Equation with Fractions as Coefficients

5 25

5 5

x

15 1 24 1x

2 4 2

Check:

The solution set of the equation is {5}.

Slide 2.3-15

Solving an Equation with Fractions as Coefficients (cont’d)

1 23 1 2

4 3x x

1 23 1 2

4 35 5

1 28 6 2

4 3

8 122

4 3

CLASSROOM EXAMPLE 7

CLASSROOM EXAMPLE 8

Solve.

Solution:

10 15 38x xx x

14 1

2

4

14 8x

2x

0.5 2 3 4.5 0.1 7x x

0.5 2 3 4.5 0.1 710 10 10x x

5 2 5 3 45 7x x

1010 13 014 8x

Slide 2.3-16

Solving an Equation with Decimals as Coefficients

Multiplying by 10 is the same as moving the decimal point one place to the right. Likewise, multiplying by 100 is the same as moving the decimal point two places to the right.

Check:

0.5 2 3 4.5 0.1 7x x

10 5 6 47 7

10 30 40 40 40

The solution set of the equation is {−2}.

5 2 5 3 45 72 2

Slide 2.3-17

Solving an Equation with Decimals as Coefficients (cont’d)CLASSROOM EXAMPLE 8

Objective 3

Solve equations with no solution or infinitely many solutions.

Slide 2.3-18

Each equation that we have solved so far has had exactly one solution. An equation with exactly one solution is a conditional equation because it is only true under certain conditions. Sometimes equations may have no solution or infinitely many solutions.

Slide 2.3-19

Solve equations with no solution or infinitely many solutions.

CLASSROOM EXAMPLE 9

Solve.

Solution:

10 10 01 10 0 0

3 10 2 4 14x x x

2 10 2 10x x 22 10 1 22 0xx x x

The variable has “disappeared.” Since the last statement is true, any real number is a solution. An equation with both sides exactly the same, is called an identity. An identity is true for all replacements of the variables. We write the solution set as {all real numbers}.

Slide 2.3-20

Solving an Equation That Has Infinitely Many Solutions

DO NOT write { 0 } as the solution set. There are infinitely many other solutions. For { 0 } to be the solution set, the last line must include a variable, such as x, and read x = 0, not 0 = 0.

CLASSROOM EXAMPLE 10

Solve.

Solution:

8 6

3 8 6 1 3x x x

3 8 6 6 3x x x 33 38 3 6xx xx

Again, the variable has “disappeared,” but this time a false statement results. When this happens in solving an equation, it indicates that the equation has no solution and is called a contradiction. Its solution set is the empty set, or null set, symbolized Ø.

DO NOT write { Ø } to represent the empty set.

Slide 2.3-21

Solving an Equation That Has No Solution

The following table summarizes the solution sets of the three types of equations presented so far.

Slide 2.3-22

Solve equations with no solution or infinitely many solutions.

Objective 4

Write expressions for two related unknown quantities.

Slide 2.3-23

Solution:

CLASSROOM EXAMPLE 11

Two numbers have a product of 36. If one of the numbers is represented by x, find an expression for the other number.

Slide 2.3-24

Translating a Phrase into an Algebraic Expression