Think about a scenario when you would use an expression and an equationin a real life application. Discuss each scenario and explain how towrite the expression and equation.

Create **two** sample equations for your peers to solve.

Module 1 Notes

Objectives:

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Simplify and evaluate expressions

Solve equations

Use and apply the order of operations

Use and apply the distributive property

Translate statements into algebra

Identify variables, coefficients, and constants

Combine like terms

Expressions and Equations

The main difference between expressions and equations is that an expression does not

have an = sign and an equation has one. An expression is an incomplete thought, and

thus we can simplify it. An equation will produce a final answer, such as x=5, and thus

we can solve it.

Variables- letters that represent a value (examples: x, y, z, a, b, c, etc.)

Coefficients- the number that is in front of a variable (examples: 3x, 2×2)

Constants- numerical values (examples: 2, 3, 4, etc.)

Example:

2×2-4x+3

This expression contains 3 terms: 2×2, -4x, and 3. The coefficients are 2 and -4. The

variable terms are x2 and x. The constant is 3.

Simplifying an expression example:

5x-4y+8x-2y-7 Circle, highlight, or identify the like terms to combine

5x+8x

Combine like terms (x’s)

-4y-2y

Combine like terms (y’s)

13x-6y-7

Final answer, note that -7 is a constant and does not have another term

to combine in this problem

Evaluating an expression example:

Let x= -3 and y= 6

7y – 4x

7(6) – 4(-3)

Substitute the values for the appropriate variables

42+12

Perform the necessary operations (recall that two negatives make a

positive)

54

Combine like terms

Like Terms

Within an expression, we have what are called like terms. These are terms, which have

‘the same last name’, and thus can be combined.

Example:

2x+3-5x

Here the 2x and -5x are like terms. Don’t forget the -, it is a big deal. The term ‘owns’

the sign in front of it.

Combine the terms together, 2x-5x= -3x

The 3 is a number (constant) and cannot be added to the -3x.

Therefore, the answer is -3x+3

Order of Operations

Another important topic that is covered in this module is the order of operations. This is

a very important concept to master, because if you aren’t working the order of

operations correctly, you may continue to struggle with the rest of the math topics.

There are several mnemonics which you may have heard for the order of operations.

PEMDAS and Please Excuse My Dear Aunt Sally are a couple of those. These are

ways to remember the proper order to simplify expressions.

P = Parentheses or any other grouping symbol, within each set of grouping symbols you

must also use the order of operations and work from the inside to the outside

E = Exponents as well as any roots

M = Multiplication and D = Division. IMPORTANT NOTE: These are equal level

operations and are always worked from left to right.

A = Addition and S = Subtraction. Again, these are equal level operations and are

worked in order from left to right.

Example:

2(8-10)2-5(2-6)2

2(8-10)2-5(2-6)2

Perform the operations in the parenthesis first

2(-2)2-5(-4)2

Simplify the exponents

2(4)-5(16)

Multiply the terms

8-80

Subtract

-72

The Order of Operations is like a recipe that must be followed. You can’t bake a cake

before you have mixed together your ingredients, so you want to make sure that you

follow the order correctly.

The Distributive Property

a * (b+c) = ab + ac

When applying the distributive property, multiply the number outside of the parentheses

to everything inside the parentheses paying attention to the signs.

Examples:

9(1+6)

9(1) + 9(6)

9+54

63

3(x+2y)

3(x) + 3(2y)

3x+6y

Absolute Value

Any number within the absolute value brackets | | will be positive. Be careful when a

negative number is outside the bracket.

Examples:

|-4|= 4

|5|= 5

|-1-5|= 6

-|5|= -5

The negative is applied after the absolute value due to the order of operations.

It is the same as multiplying -1 * 5.

-|-7|= -7 The same rule above is applied here. Multiply the negative after the absolute

value is simplified.

Translations

In algebra, we can take a written statement and convert it into an algebraic

expression/equation or vice versa, convert an expression/equation into a statement.

There are some key words to look out for which signify the different operations. With

addition and multiplication, the order is less important. However, the order needs to be

correct with subtraction and division.

Words that indicate operations

Addition (+)

sum, plus, more than, increased by, added to

Subtraction (-)

minus, less than, decreased by, from, difference, take away

Multiplication (*)

product, times, of, by

Division (÷)

divided, quotient, into, per

Equal (=)

is

Examples:

The sum of y and 3, divided by 8

(y+3)

8

The product of x and 5 more than x

x(x+5)

The product of 4 and the quantity of c and d

4(c+d)

The product of z and 8 less than y

z(y-8)

The result of p-q, divided by 2

p-q

2

Solving Equations

In an equation, you want to solve for the variable by isolating it. First move the numbers

to one side using inverse (opposite) operations. The next step is to multiply or divide to

get the variable alone. What is done to one side, do to the other to keep it balanced.

Example:

2x-4=10 Add 4 to both sides

2x-4=10

+4 +4

2x=14

/2 /2

x=7

Divide by 2

✓ Check it…Plug the answer back in the problem.

2x-4=10

2(7)-4=10

14-4=10

10=10

Both sides must equal.

Steps for Solving Equations

1. Remove parenthesis, if any

2. Combine like terms on each side of the equation

3. Move the variables on one side of the equation and the numbers on the other

side using the addition or subtraction properties of equality (inverse operations)

4. Combine like terms

5. Use the multiplication or division property of equality to solve for the variable

6. Check your answer by plugging it back into the original equation

Types of Solutions

Conditional equation- an equation that is satisfied by at least one real number but not

an identity

Identity- an equation where every real number is a solution

Inconsistent equation- an equation with no solution