Solving Linear Equations

Solving Linear Equations

Tutorial 3d

A Solution Set

Consider the different meanings of the word solution. The solution to the mystery escaped him.

The word solution here refers to an explanation. The town’s solution to its landfill problem is to

encourage recycling. Solution here refers to a method of solving a problem.

A chemist mixes two solutions to obtain a 15% acid solution.

Solution here refers to a homogeneous molecular mixture

Solution Set

In Mathematics we also have different kinds of solutions and, therefore, different kinds of solution sets.

Study the table below:These examples illustrate that a solution set may have one member, more than one member, or no members.

Equation/Inequality Solution Set

3x + 5 = 14 {3}

|x| 5 {-5 x 5}

x + 3 = x - 7 No Solution

(x + 6)(x – 3)=0 {-6, 3}

Solving:Addition & Subtraction Equations

One way to solve an equation is to get the variable alone on one side of the equal sign.

You can do this by using inverse operations, which are operations that undo one another.

Addition and subtraction are inverse operations.

You can use subtraction to undo addition and addition to undo subtraction .


Think to yourself: What is being done to the variable (x)?

A 4 is being added to the variable (x). Subtraction undoes addition therefore you should subtract a 4 on the left to get x alone on one side.

However, whatever you do to one side of an equation you must also do to the other side.

x + 4 = 7

-4 -4

Example #1: Solve the equation x + 4 = 7

x = 3

Always check your answers!x + 4 = 7; Does x = 3?3 + 4 = 7 is true therefore x = 3!



A 12 is being subtracted from the variable (x). Addition undoes subtraction, therefore you should add a 12 on the left to get x alone on one side.


x - 12 = 20

+12 +12

Example #2: Solve the equation x - 12 = 20

x = 32

Always check your answers! x - 12 = 20; Does x = 32?32 - 12 = 20 is true therefore x = 32!

Problem Solving:

Think to yourself: What is being done to the variable (w)?

A 135 is being added to the variable (w). Subtraction undoes addition therefore you should subtract a 135 on the left to get w alone on one side.


w + 135 = 193.7

-135 -135

A veterinary assistant holds a dog and steps on a scale. The scale reads 193.7 lb. Alone, the assistant weighs 135 lb. To find the weight of the dog, solve the equation w + 135 = 193.7

w = 58.7

Always check your answers! w + 135 = 193.7; Does w = 58.7?58.7 + 135 = 193.7 is true!

The dog weighs 58.7 lb.

Solving:Multiplication & Division Equations

Multiplication and division are inverse operations.

You can use division to undo multiplication and multiplication to undo division.



A 5 is being multiplied to the variable (x). Division undoes multiplication, therefore you should divide a 5 to the left side to get x alone on that side.


5x = 35

Example #1: Solve the equation 5x = 35

x = 7

Always check your answers! 5x = 35; Does x = 7?5•7 = 35 is true therefore x = 7 !

5 5

1 7

1 1


Think to yourself: What is being done to the variable (r)?

A 6 is being divided into the variable (r). Multiplication undoes division, therefore you should multiply a 6 to the left side to get r alone on that side.


Example #2: Solve the equation

r = 24

Always check your answers!

6• • 6

46

r

46

r1

1

? Does ; 2446

rr

! therefore true is 2446

24 r

5

6


Think to yourself: What is being done to the variable (r)?

A 5/6 is being multiplied to the variable (r). Multiplying by the reciprocal will eliminate the fraction, therefore you should multiply a 6/5 to the left side to get r alone on that side.


Example #2: Solve the equation

r = 108

Always check your answers!

906

5r

906

5r

1

1

?108 Does ; 906

5 rr

!108 therefore trueis 901086

5 r

5

6

1

1

1

18

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Solving Linear Equations