Linear Equations, Inequalities, and Absolute Value - Solving Linear Equations METHOD #1 – Inspection Some equations are easy to solve. Ones that have one

Linear Equations, Inequalities, and Absolute Value

- Solving Linear Equations

METHOD #1 – Inspection

Some equations are easy to solve. Ones that have one operation involved and you can “talk” or reason you way thru the answer.

An equation is solved ONLY when the unknown has a coefficient of one.





EXAMPLE # 1 : Solve for “a” :

125 a







125 a

Something added to 5 equals 12. You could start testing numbers until you get the answer or just “know” the answer will be seven.







153 a

Something times ( - 3 ) equals 15. You could start testing numbers until you get the answer or just “know” the answer will be five.




METHOD #2 – Isolation

This method uses opposite operations. You will “isolate” the variable by getting it by itself on one side of the equation and the answer on the other side.






OPPOSITE OPERATIONS : ADD - SUBTRACT

MULTIPLY - DIVIDE






OPPOSITE OPERATIONS : ADD - SUBTRACT

MULTIPLY - DIVIDE

STEPS : 1. On which side of the equation is my unknown ?

2. If the unknown appears on both sides, you’ll need to get it on one side

3. Move all constants first

4. Move all coefficients last




EXAMPLE 1 : Solve

2082 x



EXAMPLE 1 : Solve

2082 x The left side = the right side.

Acts like a seesaw. If I add 10 lbs to one side, I have to add 10 lbs to the other side to keep it balanced.





Variable is on left side, so I will move things from the left side to the right side…



EXAMPLE 1 : Solve

2082 x





88 +8 needs to move, so I will subtract 8 from both sides…



EXAMPLE 1 : Solve

2082 x





88

12 2 x

You can see that now we have simplified the left side and our unknown is almost by itself.

Remember, to be by itself, it needs to have a coefficient of one…



EXAMPLE 1 : Solve

2082 x





88

2

12

2

2

x

Our unknown is being multiplied by 2, so we will divide BOTH sides by 2…



EXAMPLE 1 : Solve

2082 x





88

2

12

2

2

x

Our unknown is being multiplied by 2, so we will divide BOTH sides by 2…

6x ANSWER



EXAMPLE 1 : Solve

2082 x





88

2

12

2

2

x

Let’s check this answer back into the original equation…6x

2020

20812

20862

2082

x



EXAMPLE 2 : Solve

24

14 x

Variable is on left side, so I will move things from the left side to the right side…



EXAMPLE 2 : Solve

24

14 x

Place a box around your variable, its coefficient, and any sign that is in front…

Another way to attack the first step…



EXAMPLE 2 : Solve

24

14 x

What do you see outside the box on the same side of the equation ?…




EXAMPLE 2 : Solve

24

14 x

What do you see outside the box on the same side of the equation ?…


( - 4 )…so ADD 4 to both sides…



EXAMPLE 2 : Solve

24

14 x

Add 4 to both sides…4 4

64

1 x



EXAMPLE 2 : Solve

24

14 x

When the coefficient of your unknown is a fraction, multiply both sides by the reciprocal ( flip your fraction )…

4 4

64

1 x

1

46

4

1

1

4x



EXAMPLE 2 : Solve

24

14 x

When the coefficient of your unknown is a fraction, multiply both sides by the reciprocal ( flip your fraction )…

4 4

64

1 x

1

46

4

1

1

4x

24x



EXAMPLE 3 : Solve

9315 x

Variable is on right side, so I will move things from the right side to the left side…



EXAMPLE 3 : Solve

9315 xPlace a box around your variable, its coefficient, and any sign that is in front…



EXAMPLE 3 : Solve

9315 x What do you see outside the box on the same side of the equation ?…



EXAMPLE 3 : Solve


( + 9 )…so subtract 9 from both sides…

9 9



EXAMPLE 3 : Solve


( + 9 )…so subtract 9 from both sides…

9 9

x324



EXAMPLE 3 : Solve

9315 x

Unknown is being multiplied by ( - 3 ) so we will divide both sides by ( - 3 )

9 9

x324



EXAMPLE 3 : Solve

9315 x


9 9

3

3

3

24

x



EXAMPLE 3 : Solve

9315 x


9 9

3

3

3

24

x

x8



EXAMPLE 4 : Solve

10776 xx

OK, in this example the variable appears on both sides of the equation.



EXAMPLE 4 : Solve

10776 xx


We’ll need to get it on one side so we need to move one of them.



EXAMPLE 4 : Solve

10776 xx


We’ll need to get it on one side so we need to move one of them.

I’m going to move the 7x from the right side to the left side…



EXAMPLE 4 : Solve

10776 xxMove the 7x from the right side to the left side…xx 7 7



EXAMPLE 4 : Solve

10776 xxMove the 7x from the right side to the left side…xx 7 7

107 x



EXAMPLE 4 : Solve

10776 xx

Place a box around your unknown and any coefficient and signs…

xx 7 7

107 x



EXAMPLE 4 : Solve

10776 xx

You see +7, so subtract seven from both sides…

xx 7 7

107 x



EXAMPLE 4 : Solve

10776 xx

You see +7, so subtract seven from both sides…

xx 7 7

107 x

77

3 x



EXAMPLE 4 : Solve

10776 xx

The solution MUST be positive so divide both sides by ( - 1 )…

xx 7 7

107 x

77

3 x



EXAMPLE 4 : Solve

10776 xx

The solution MUST be positive so divide both sides by ( - 1 )…

xx 7 7

107 x

77

1

3

1

x



EXAMPLE 4 : Solve

10776 xx

ANSWER

xx 7 7

107 x

77

1

3

1

x

3x



EXAMPLE 5 : Solve

11253 xx



EXAMPLE 5 : Solve

11253 xxxx 3 3 Get the variables on one side…



EXAMPLE 5 : Solve

11253 xxxx 3 3 Get the variables on one side…

1155 x



EXAMPLE 5 : Solve

11253 xxxx 3 3

Add 11 to both sides…

1155 x11 11

x516



EXAMPLE 5 : Solve

11253 xxxx 3 3

Divide both sides by 5…

1155 x11 11

5

5

5

16 x



EXAMPLE 5 : Solve

11253 xxxx 3 3

You might get a fraction for an answer. Leave improper fractions as improper…

1155 x11 11

5

5

5

16 x

x5

16



EXAMPLE 6 : Solve



EXAMPLE 6 : Solve

1st - distribute



EXAMPLE 6 : Solve

Subtract 20 from both sides



EXAMPLE 6 : Solve

Subtract 20 from both sides



EXAMPLE 6 : Solve



EXAMPLE 6 : Solve



EXAMPLE 6 : Solve



EXAMPLE 6 : Solve

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Linear Equations, Inequalities, and Absolute Value - Solving Linear Equations METHOD #1 – Inspection Some equations are easy to solve. Ones that have one