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Intro to Algebraic Expressions
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 An algebraic expression is a number sentence with an unknown number.
 How to simplify algebraic expressions by combining like terms.
 And we'll see how this knowledge can help us count savings, order pizza, and even study earthquakes!

Discussion Questions

Before VideoWhat number is missing? How do you know? 5+ ___ =12ANSWER

7; count on 7 units from 5, or subtract 5 from 12.

14; add 3 and 11 to get 14.

question mark, empty box, letter such as x

The last expression includes subtraction, while the first two use addition. The last two expressions include unknowns, while the first expression has only numbers.

5+93+16=143+16=11+16=27


After VideoGive examples of numerical expressions and algebraic expressions.ANSWER

An expression without a variable, such as 18+3, and with a variable, such as m4.

An equation has two expressions set equal, showing that they have the same value.

Five: 4x,2, 8, 3y, 9

5 and 2 are coefficients, and 9 is a constant.

3x+17; 35



Vocabulary

Algebraic expression
DEFINE
A number sentence with an unknown number.

Simplify
DEFINE
To make as simple as possible, such as by combining like terms to produce a more compact expression.

Expression
DEFINE
An incomplete number sentence (without an equal sign).

Equation
DEFINE
A complete number sentence stating that two expressions are equal to each other.

Numerical expression
DEFINE
An expression that does not have any variables.

Variable
DEFINE
An unknown amount represented by a letter.

Terms
DEFINE
Parts of an expression that are separated by addition or subtraction.

Coefficient
DEFINE
The number that is multiplied by a variable in a term.

Constant
DEFINE
A numerical term (it has no variable).

Like terms
DEFINE
Terms that contain the same variable, if any, and the same exponent, if any (for example, 3x and 4x, 5 and 8, 8y2 and 6y2).

Algebraic expression
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageWHAT ARE ALGEBRAIC EXPRESSIONS?An algebraic expression is an expression that includes a variable. A variable is a letter that we use to represent an unknown value. For example, 3x + 2 is an algebraic expression because of the variable, x. You can use algebraic expressions to represent situations where there is some information that you do not know.
To better understand algebraic expressions…
WHAT ARE ALGEBRAIC EXPRESSIONS?. An algebraic expression is an expression that includes a variable. A variable is a letter that we use to represent an unknown value. For example, 3x + 2 is an algebraic expression because of the variable, x. You can use algebraic expressions to represent situations where there is some information that you do not know. To better understand algebraic expressions…LET’S BREAK IT DOWN!
An Algebraic Expression Contains an Unknown Amount
An algebraic expression can include numbers, operations, and variables. Let’s say a piggy bank has at least $4 in it, but there’s some unknown amount more than that in the bank. You can use the algebraic expression 4+x to represent the total amount of money in the piggy bank, where x represents the unknown amount. If the unknown amount turned out to be $6, then you could evaluate the expression 4+x by substituting 6 for x. The result is 4 + 6 = 10, so there is $10 in the piggy bank. Try this one yourself: Evaluate the algebraic expression x – 8 when x = 15.
An Algebraic Expression Contains an Unknown Amount An algebraic expression can include numbers, operations, and variables. Let’s say a piggy bank has at least $4 in it, but there’s some unknown amount more than that in the bank. You can use the algebraic expression 4+x to represent the total amount of money in the piggy bank, where x represents the unknown amount. If the unknown amount turned out to be $6, then you could evaluate the expression 4+x by substituting 6 for x. The result is 4 + 6 = 10, so there is $10 in the piggy bank. Try this one yourself: Evaluate the algebraic expression x – 8 when x = 15.There are Different Types of Expressions and Equations
A numerical expression does not contain variables. You can evaluate basic numerical expressions, such as 5 + 8 or 3 × 6, to a single number. This is called “simplifying.” An algebraic expression includes at least one unknown (variable). In earlier grades, you might have represented the unknown by an empty square, a fillintheblank line, or a question mark. It’s common for the unknown to be represented with the variable x, but it could just as easily be some other letter, such as y or b. Expressions can contain different operations, such as x+4 for “four more than x,” 8z for “z less than eight,” 2.5y for “two and one half times y,” and m3 for “m divided by three.” When two expressions have the same value, they can be set equal to each other. This creates an algebraic equation. For example, x+5=2x is an equation stating that the value of x+5 is the same as the value of 2x. Try this one yourself: Write an algebraic expression for “the sum of x and 10.”
There are Different Types of Expressions and Equations A numerical expression does not contain variables. You can evaluate basic numerical expressions, such as 5 + 8 or 3 × 6, to a single number. This is called “simplifying.” An algebraic expression includes at least one unknown (variable). In earlier grades, you might have represented the unknown by an empty square, a fillintheblank line, or a question mark. It’s common for the unknown to be represented with the variable x, but it could just as easily be some other letter, such as y or b. Expressions can contain different operations, such as x+4 for “four more than x,” 8z for “z less than eight,” 2.5y for “two and one half times y,” and m3 for “m divided by three.” When two expressions have the same value, they can be set equal to each other. This creates an algebraic equation. For example, x+5=2x is an equation stating that the value of x+5 is the same as the value of 2x. Try this one yourself: Write an algebraic expression for “the sum of x and 10.”Expressions Can Have Different Parts
Sometimes expressions contain more than one operation. Parts of an expression that are separated by addition or subtraction signs are called “terms.” The expression 3x+5 represents “three times x plus 5”. That is, 3 is multiplied by an unknown number and then five is added to the result. This expression has two terms: 3x and 5. Expressions can be longer and have more terms. The expression 8x+73x+8 has four terms: 8x, 7, 3x, and 8. Some of these terms contain the variable x, but others do not. When a term contains a variable, as in the terms 8x and 3x, the number multiplied by the variable is called the “coefficient”. When a term has no variable, as in the terms 7 and 8, the number is called a “constant”. The expression 2x+5 has two terms, 2x and 5. The first term has coefficient 2 and variable x. The second term has constant 5. Try this one yourself: Identify the terms, coefficients, and constants in the expression 4x+6y+9.
Expressions Can Have Different Parts Sometimes expressions contain more than one operation. Parts of an expression that are separated by addition or subtraction signs are called “terms.” The expression 3x+5 represents “three times x plus 5”. That is, 3 is multiplied by an unknown number and then five is added to the result. This expression has two terms: 3x and 5. Expressions can be longer and have more terms. The expression 8x+73x+8 has four terms: 8x, 7, 3x, and 8. Some of these terms contain the variable x, but others do not. When a term contains a variable, as in the terms 8x and 3x, the number multiplied by the variable is called the “coefficient”. When a term has no variable, as in the terms 7 and 8, the number is called a “constant”. The expression 2x+5 has two terms, 2x and 5. The first term has coefficient 2 and variable x. The second term has constant 5. Try this one yourself: Identify the terms, coefficients, and constants in the expression 4x+6y+9.You Can Combine Like Terms
When an expression contains like terms, we can combine the terms to simplify the expression. When terms have no variables, this is obvious. The expression 9+5 has two constant terms, so simplifies to 14. A more complicated expression such as 8x+2x+4+9 has two xterms and two constant terms. We can combine like terms with like. So, 8x + 2x = 10x, and 4 + 9 = 13. So 8x + 2x + 4 + 9 simplifies to 10x+13. When the expression includes subtraction, be careful to use the subtraction sign with the number immediately after it. The expression 5x3+2x is the same as (5x + 2x) – 3 = 7x – 3. To simplify an expression that contains more than one variable, we still only combine terms with the same variable. For example, simplify 2x+5x+y2. Combine like terms: (2x – x) + y + (5 – 2) = x + y + 3. Try this one yourself: Simplify the expression: 3x+92+x+5y6.
You Can Combine Like Terms When an expression contains like terms, we can combine the terms to simplify the expression. When terms have no variables, this is obvious. The expression 9+5 has two constant terms, so simplifies to 14. A more complicated expression such as 8x+2x+4+9 has two xterms and two constant terms. We can combine like terms with like. So, 8x + 2x = 10x, and 4 + 9 = 13. So 8x + 2x + 4 + 9 simplifies to 10x+13. When the expression includes subtraction, be careful to use the subtraction sign with the number immediately after it. The expression 5x3+2x is the same as (5x + 2x) – 3 = 7x – 3. To simplify an expression that contains more than one variable, we still only combine terms with the same variable. For example, simplify 2x+5x+y2. Combine like terms: (2x – x) + y + (5 – 2) = x + y + 3. Try this one yourself: Simplify the expression: 3x+92+x+5y6.Algebraic Expressions Can Represent Buying Things
Let’s say you're ordering pizza. You order 2 pizzas for an unknown number of dollars each. You can represent this situation with the expression 2x, where x is the cost of each pizza. If you also buy a drink for $3, the total cost is 2x+3. If it turns out that each pizza costs $10, you can evaluate the expression for x=10. Calculate (2×10)+3 for a total of $23. Try this one yourself: Three packs of gum each started with the same number of pieces of gum, x, and you already chewed 5 pieces. Write an algebraic expression for the number of pieces of gum left in the packs. Then evaluate your expression, assuming there were 8 pieces of gum in each pack before you chewed any of it.
Algebraic Expressions Can Represent Buying Things Let’s say you're ordering pizza. You order 2 pizzas for an unknown number of dollars each. You can represent this situation with the expression 2x, where x is the cost of each pizza. If you also buy a drink for $3, the total cost is 2x+3. If it turns out that each pizza costs $10, you can evaluate the expression for x=10. Calculate (2×10)+3 for a total of $23. Try this one yourself: Three packs of gum each started with the same number of pieces of gum, x, and you already chewed 5 pieces. Write an algebraic expression for the number of pieces of gum left in the packs. Then evaluate your expression, assuming there were 8 pieces of gum in each pack before you chewed any of it.Careers That Use Algebraic Expressions
A seismologist studies earthquakes. Seismologists set up algebraic expressions to determine the amount of damage an earthquake could do to a city based on the earthquake’s size and distance from the city. Business owners who sell clothes set up algebraic expressions to find out how much money they will make based on the number of items they sell and the price of those items. Filmmakers who make movies use algebraic expressions to predict how many movie tickets they must sell to make a profit on their films.
Careers That Use Algebraic Expressions A seismologist studies earthquakes. Seismologists set up algebraic expressions to determine the amount of damage an earthquake could do to a city based on the earthquake’s size and distance from the city. Business owners who sell clothes set up algebraic expressions to find out how much money they will make based on the number of items they sell and the price of those items. Filmmakers who make movies use algebraic expressions to predict how many movie tickets they must sell to make a profit on their films. 
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Circle the numerical expressions, and box the algebraic expressions. Leave any equations unmarked. 3x 15 y+7=24 [ggfrac]1/6[/ggfrac] [ggfrac]n/2[/ggfrac]
Identify each term, coefficient, variable, and constant in the expression below. 5x+43x+y6 Terms: ______________ Coefficients: _________ Variables: ____________ Constants: ___________
Simplify the expression 8n+73n+2. Then evaluate the simplified expression for n=4.
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