Instructor: Manish NarayanProblem Solving

Given: 5 = 5 A number can be added to both sides of the equation without

changing the equality: 5 + 20 = 5 + 20 (note that the same number must be added to

both sides) 25 = 25 A number can be subtracted from both sides of the equation

without changing the equality: 5 – 3 = 5 – 3 2 = 2 A number can be multiplied to both sides of the equation without

changing the equality: 5*10 = 5*10 50 = 50 A number can be divided into both sides of the equation without

changing the equality: 5/2 = 5/2 2.5 = 2.5

Variables: a variable can be assigned different values and these values are assigned via the substitution property.

Examples of Variables: x, y, z (any letter can be defined as a variable) x can be assigned the value of 4 so that x = 4

Constants: constants are numbers that do not change such as pi = 3.1415. Pick any number and it is a constant as the value itself does not change. (Example: 4, 9, 100, etc.)

In order to solve an equation for a specified variable, the variable must be isolated to one side of the equal sign:

For example: x + 5 = 12 The variable is x while the numbers 5 and

12 are constants. The goal is to isolate x by itself.

Before isolating variables some rules must be defined in the following slide.

First the inverse operations must be defined as follows:

The opposite of Addition is Subtraction and the opposite of Subtraction is Addition

The opposite of Multiplication is Division and the opposite of Division is Multiplication.

Next the order of operations must be defined as follows (remember this is different from PEMDAS):

1. Get rid of addition and subtraction first 2. Get rid of multiplication and division

second Examples provided in the following slides

Solve the following equation for x: x + 4 = 10 Is x by itself? NO What operation is attached to x? addition Opposite of addition is subtraction so we

subtract 4 from both sides of the equation: x + 4 = 10 - 4 = -4 x = 6 Notice how the 4’s cancel out

Solve the following equation for x: x - 3 = 12 Is x by itself? NO What operation is attached to x?

subtraction Opposite of subtraction is addition so we

add 3 to both sides of the equation: x - 3 = 12 + 3 = +3 x = 15 Notice how the 3’s cancel out

Solve the following equation for x: 4x = 12 Is x by itself? NO What operation is attached to x?

multiplication Opposite of multiplication is division so we

divide 3 into both sides of the equation: 4x/4 = 12/4 x = 3 Notice how the 4’s divide to give 1 on the

left side.

Solve the following equation for x: x/7 = 10 Is x by itself? NO What operation is attached to x? Division Opposite of division is multiplication so we

multiply 7 to both sides of the equation: 7*x/7 = 10*7 x = 70 Notice how the 7’s divide to give 1 on the

left side.

Sometimes multiple steps are required to isolate the specified variable as follows:

Solve the equation for x: 2x – 5 = 11 Is x by itself? NO What operations are attached to x? multiplication and

subtraction What operation comes first? Subtraction Opposite of subtraction is addition so we add 5 to both sides of

the equation: 2x – 5 = 11 + 5 = +5 2x = 16 What operation comes second? Multiplication Opposite of multiplication is division so we divide 2 into both

sides of the equation: 2x/2 = 16/2 x = 8

Sometimes we deal must deal with many variables at once…don’t worry just follow our simple rules:

Example: Solve the equation for x: xy = w Is x by itself? NO What operation is attached to x? Multiplication Opposite of multiplication is division so we

divide y into both sides of the equation: xy/y = w/y x = w/y notice how the y’s on the left side

cancel out.

If I have 2 apples and you give me 3 apples, I now have 5 apples or in terms of an equation:

2 apples + 3 apples = 5 apples Only like terms can be added or subtracted together 2 apples + 3 oranges = 2 apples + 3 oranges We are unable to add apples and oranges as they are unlike

terms. This same logic holds true for variables as well. 2x + 3x = 5x (if I have 2 x’s and you give me 3 x’s then I now

have 5 x’s) Some examples of like and unlike terms: x, x (like terms) x^2, x (unlike terms as they have different exponents) xy, xy (like terms) xw, xw^2 (unlike terms as the x’s are the same but the w’s have

different exponents) 4, 7 (like terms as they are both “constants”, note that the

numbers are different but are of the same type and can be added or subtracted)

Simplify the following expression (good idea to underline the like terms):

3x + 8 + 5x – 2 First underline the like terms: 3x + 8 + 5x – 2 8x + 6 (answer) Note answer can also be written as: 6 +

8x