1.1 Algebraic Expressions and the Order of Operations

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Using Variables:

Using Variables Lesson 1.1


Definitions A variable is a symbol, usually a letter, that represents one or more numbers. Remember, the word “variable” means that something can change. In math, the value of the variable can change. X might have a value of 13 in one problem, but a value of -234 in a different problem. An algebraic expression is a mathematical phrase that can include numbers, variables, and operations.

Writing Algebraic Expressions:

Writing Algebraic Expressions This is nothing more than translating words to a mathematical expression. Certain key phrases indicate the operation: more than, less than, increased by, decreased by, fewer, product, etc. Make sure your numbers are in the correct order with the correct operation!

Writing Expressions Example:

Writing Expressions Example Write as an algebraic expression: 6 more than a number. “A number” means to choose a variable. I’m going to use x. (You can choose whatever letter you like, unless you are given specific directions.) 6 more than x “More than” means to add something, so we’re adding 6 and a number. However, there is one catch: any time you see “more than”, “less than”, “fewer than”, you need to be sure to switch the order. X + 6

Writing Example 2:

Writing Example 2 The product of 5 and a number . Again, “a number” means to choose a letter. I will once again choose x . The product of 5 and x . “Product” means to multiply. Any time you have to multiply a number and a letter together, just write them side-by-side, with no space in between. The number goes first. 5 x

Why do we flip when we see “than”?:

Why do we flip when we see “than”? In expressions like these, “than” is a comparison. However, it’s not comparing things in the “normal” order. Consider this. Marianne’s age is 8 years less than Mr. Scofield’s . If we just write the expression normally, we’d have 8 - x

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So, knowing that Mr. Scofield is 36, if we plug that value in for x, we get Marianne’s age. 8 – x = 8 – 36 = -28 Therefore, Marianne is -28 years old. Houston, we have a problem!!!!! Why did it go wrong? It’s very simple: the 8 is the number of years we have to remove from Mr. Scofield’s age. However, that’s not what we wrote. We have to account for the change in wording, so we flip.

Translating Algebraic Expressions:

Translating Algebraic Expressions Going backwards now: you’re given the mathematical sentence, and you must write it in words. Avoid using the words “add”, “subtract”, “multiply”, and “divide”. You’re NOT reading the problem out loud. You’re describing it. “Add”, “subtract”, “multiply”, and “divide” are used for reading problems. Just take the expression, determine what happened (did you add, multiply, etc ), and then find a way to say it. Use Nerdspeak , not English.

Translating Examples:

Translating Examples 12 ÷ x You’re dividing here, so you could use the word “quotient”: “The quotient of 12 and a number.” N – 5 “A number reduced by 5” Or “The difference of a number and 5.” Or “5 less than a number” NOT N minus 5.

Defining a variable and writing an expression:

Defining a variable and writing an expression If you are told to “define a variable and write an expression”, it’s extremely easy to do. For example: “ a number increased by 150” First, define the variable. This literally means to tell what the variable stands for. “Let x equal a number” Next, write the expression. “x + 150”

Order of Operations:

Order of Operations Frequently known by the mnemonic device “Please Excuse My Dear Aunt Sally, she Left to Right .” (PEMDAS) This gives us the steps to follow in evaluating mathematical expressions.

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P: Parentheses E: Exponents M: Multiply OR D: Divide A: Add OR S: Subtract L: Left TO R: Right


WARNING!!!! Note that it does NOT say “multiply, then divide” or “Add, then subtract”. It says “Multiply OR divide”, and “Add OR subtract”, from left to right. In other words, do the operation you see first, going from left to right. If you see a division sign before a multiplication sign, you MUST divide first. If you see a subtraction sign before an addition sign, you MUST subtract first.

PEMDAS Example:

PEMDAS Example To use PEMDAS, simply start at the left side of the expression and follow each step. 36 ÷ 6 · 2 + 5 6 · 2 + 5 1 2 + 5 17

Evaluating Expressions:

Evaluating Expressions “Evaluate” simply means “find the value”, or “find the answer.” Just plug in the given values and follow the order of operations.

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Evaluate 3x + 2y – 10z for x = 2, y = -6, z = 10 3(2) + 2(-6) -10(10) 6 + 2(-6) -10(10) 6 + -12 – 10(10) 6 + -12 - 100 -6 - 100 -6 + -100 -106

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Evaluate when g = -3

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