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# Patterns and Expressions

## Write variable expressions to represent word problems.

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Patterns and Expressions

Jeremy read that degrees Celsius converted to degrees Fahrenheit is "the sum of 32 and 95\begin{align*}\frac{9}{5}\end{align*} times the temperature in degrees Celsius." However, he's not sure how to convert this into an algebraic expression. What do you think an equivalent algebraic expression would be?

### Writing Algebraic Expressions

In mathematics, especially in algebra, we look for patterns in the numbers that we see. Using mathematical verbs and variables, expressions can be written to describe a pattern. Recall that an algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations. We can describe patterns using phrases as well, and we want to be able to translate these phrases into algebraic expressions.

We can use x\begin{align*}x\end{align*} to represent the number of shirts. For the cost, we have a fixed $100 charge and then$7 times the number of shirts printed. This can be expressed as 100+7x\begin{align*} 100+7x\end{align*}. For the revenue, we have 15 times the number of shirts sold, or 15x\begin{align*} 15x\end{align*}. ### Review For exercises 1 – 15, translate the English phrase into an algebraic expression. For the exercises without a stated variable, choose a letter to represent the unknown quantity. 1. Sixteen more than a number 2. The quotient of h\begin{align*}h\end{align*} and 8 3. Forty-two less than y\begin{align*}y\end{align*} 4. The product of k\begin{align*}k\end{align*} and three 5. The sum of g\begin{align*}g\end{align*} and 7\begin{align*}-7\end{align*} 6. r\begin{align*}r\end{align*} minus 5.8 7. 6 more than 5 times a number 8. 6 divided by a number minus 12 9. A number divided by 11\begin{align*}-11\end{align*} 10. 27 less than a number times four 11. The quotient of 9.6 and m\begin{align*}m\end{align*} 12. 2 less than 10 times a number 13. The quotient of d\begin{align*}d\end{align*} and five times s\begin{align*}s\end{align*} 14. 35 less than x\begin{align*}x\end{align*} 15. The product of 6, 9\begin{align*}-9\end{align*}, and u\begin{align*}u\end{align*} In exercises 16 – 24, write an English phrase for each algebraic expression 1. J9\begin{align*}J - 9\end{align*} 2. n14\begin{align*}\frac{n}{14}\end{align*} 3. 17a\begin{align*}17-a\end{align*} 4. 3l16\begin{align*}3l-16\end{align*} 5. 12(h)(b)\begin{align*}\frac{1}{2} (h)(b)\end{align*} 6. b3+z2\begin{align*}\frac{b}{3} + \frac{z}{2}\end{align*} 7. 4.72f\begin{align*}4.7-2f\end{align*} 8. 5.8+k\begin{align*}5.8 + k\end{align*} 9. 2l+2w\begin{align*}2l+2w\end{align*} In exercises 25 – 28, define a variable to represent the unknown quantity and write an expression to describe the situation. 1. The unit cost represents the quotient of the total cost and number of items purchased. Write an expression to represent the unit cost of the following: The total cost is14.50 for n\begin{align*}n\end{align*} objects.
2. The area of a square is the side length squared.
3. The total length of ribbon needed to make dance outfits is 15 times the number of outfits.
4. What is the remaining amount of chocolate squares if you started with 16 and have eaten some?

Use your sense of variables and operations to answer the following questions.

1. Describe a real-world situation that can be represented by h+9\begin{align*}h + 9\end{align*}.
2. What is the difference between 7m\begin{align*}\frac{7}{m}\end{align*} and m7\begin{align*}\frac{m}{7}\end{align*}?

To see the Review answers, open this PDF file and look for section 1.6.

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### Vocabulary Language: English Spanish

algebraic expression

An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations.

$\therefore$

The symbol $\therefore$ means "therefore" or "because of this."

Algebraic Equation

An algebraic equation contains numbers, variables, operations, and an equals sign.

domain

The domain of a function is the set of $x$-values for which the function is defined.

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.

horizontal axis

The horizontal axis is also referred to as the $x$-axis of a coordinate graph. By convention, we graph the input variable on the $x$-axis.

Range

The range of a function is the set of $y$ values for which the function is defined.

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

vertical axis

The vertical axis is also referred to as the $y$-axis of a coordinate graph. By convention, we graph the output variable on the $y$-axis.