Section 1
The Language of Algebra
A _______ is a letter that stands for a number.
Variables and/or numbers can be combined with mathematical operations to create algebraic _______.
An __________ is a mathematical sentence that contains an $=$ symbol.
Words such as is, was, gives, and yields translate to an _________ symbol.
Phrases such as increased by and more than are used to indicate the operation of ________ Phrases such as decreased by and less than are used to indicate the operation of
Equations that express a mathematical relationship between two or more variables are called ________.
Classify each of the following as an expression or an equation.a. $6 x-5$b. $P=a+b+c$c. $\frac{s+9 t}{8}$d. $\quad \sqrt{2 w^{2}}$
What arithmetic operations does the expression $\frac{40-8 n}{5}$ contain? What variable does it contain?
Translate each verbal model into a mathematical model.a. 7, times, the age of a dog in years, gives, the dog's equivalent human age.b. The take-home pay, will be, $\$ 2,500$, minus, any deductions.
a. The number of decades, is, the number of years, divided by 10.b. The cost of dining out, equals, the cost of the meal, plus $\$ 15$ for parking.
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.7 minutes less than the time walking
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.50 meters less than the height
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.$54 \%$ of the enrollment
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.$12 \%$ of a number
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.35 dollars more than twice the price
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.0.25 ounces more than twice the weight
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.one-half of the sum of a number and 4
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.one-third of the sum of the length and width
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.three-fourths of the pressure
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.nine-sixteenths of the thickness
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.the ratio of the number of games won and games played
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.the ratio of the number of vacation days and work days
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.950 increased by $10 \%$ of the volume
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.100 less than triple the attendance $a$
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.$w$ reduced by 500
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.$8,000$ split $n$ equal ways
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.150 feet per $m$ minutes
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.exceeds the cost by $25,000$
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.7 times the total of $77, h,$ and 88
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.decrease a number by $-1$
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.triple the number of waiters
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.680 fewer than the population
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.the product of $d$ and $4,$ decreased by 15
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.the quotient of the base and twice the height
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.$95 \%$ of the sum of 200 and the tonnage
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.14 less than eleven-ninths of a number
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.one-hundredth of the distance
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.double the difference of a number and 18
Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen.The cost each semester is the sum of $\$ 13$ times the number of units taken and a student services fee of $\$ 24$
Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen.The yearly salary is $\$ 25,000$ plus $\$ 75$ times the number of years of experience.
Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen.The quotient of the number of clients and seventy-five gives the number of social workers needed.
Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen.The difference between 500 and the number of people in a theater gives the number of unsold tickets.
Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen.Each test score was increased by 15 points to give a new adjusted test score.
Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen.The weight of a super-size order of French fries is twice that of a regular-size order.
Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen.The product of the number of boxes of crayons in a case and 12 gives the number of crayons in a case.
Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen.The perimeter of an equilateral triangle can be found by tripling the length of one of its sides.
Write a mathematical model for each situation. Answers may vary depending on the variables chosen.Taxes. $\quad$ A married couple has decided to split the money equally when they receive their federal income tax refund. Furthermore, the husband is going to donate $\$ 75$ of his share to charity. Describe the relationship between the amount of money that the husband will keep and the amount of the couple's refund.
Write a mathematical model for each situation. Answers may vary depending on the variables chosen.Copiers. $\quad$ A business is going to rent a copy machine. Under the rental agreement, the company is charged $\$ 105$ per month and 3 $\boldsymbol{\alpha}$ for every copy that is made. Describe the relationship between the monthly copier expense and the number of copies made.
Write a mathematical model for each situation. Answers may vary depending on the variables chosen.Bottled Water. $\quad$ A driver left a production plant with 300 fivegallon bottles of drinking water on his truck. His delivery route consisted of office buildings, each of which was to receive 6 bottles of water. Describe the relationship between the number of bottles of water left on his truck and the number of stops that he has made.
Write a mathematical model for each situation. Answers may vary depending on the variables chosen.Collectibles. A woman inherited 9 antique dolls. She decided to add to her collection by purchasing two more dolls each month. Describe the relationship between the number of antique dolls in her collection and the number of months since she began to purchase them.
Use the given equation to complete each table.$$c=\frac{p}{12}$$table cant copy
Use the given equation to complete each table.$$y=100 c$$(table cant copy)
Use the given equation to complete each table.$$n=22.44-K$$(table cant copy)
Use the given equation to complete each table.$$y=2 x+15$$(table cant copy)
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.a. $s$ subtracted from $S$b. $S$ subtracted from $s$
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.a. the product of 4 and $d,$ decreased by 15b. the product of 4 and $d$ decreased by 15
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.a. the absolute value of the difference of $a$ and 2b. the difference of the absolute value of $a$ and 2
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.a. the quotient of a number and 6 increased by the numberb. the quotient of a number and $6,$ increased by the number
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.a. 6 miles more than $15.5 \%$ of the altitudeb. $15.5 \%$ of 6 miles more than the altitude
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.a. twice the sum of the tax and 200b. the sum of twice the tax and 200
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.a. the square of 14 less than a numberb. 14 less than the square of a number
Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen.a. double the cube of a numberb. the cube of double a number
Production Planning. Suppose $r$ towel racks like that shown below are to be manufactured. Complete the two equations that could be used to order the necessary number of bar holders $b$ and wood screws $s$$$b=r \quad s=r$$
Staircases. $\quad$ A builder is going to construct $h$ new homes, each of which will have a staircase as shown. Complete the four equations that could be used to order the necessary number of balusters $b,$ handrails $r,$ posts $p,$ and treads $t$ for the entire project.$$\begin{aligned}&b=h\\&r=h\\&p=h\\&t=h\end{aligned}$$
When discussing weight management with their patients, dietitians stress the importance of limiting calorie intake and increasing physical activity. One method dietitians use to determine if a patient has a daily calorie surplus (or deficit) is to subtract the patient's daily resting metabolic rate and the calories that he/she burns during physical activity that day from the calories he/she ingests that day. Translate this verbal model into a mathematical model.
a. Write a verbal model that states the relationship between the cost $C$ of renting the carpet cleaning system and the number of hours $h$ it is rented.b. Translate the verbal model written in part (a) to a mathematical model.c. Use your result from part (b) to complete the table.
Use the data in each table to find an equation that mathematically describes the relationship between the two quantities.$$\begin{array}{|c|c|}\hline \text { Tower height (ft) } & \text { Height of base (ft) } \\\hline 15.5 & 5.5 \\\hline 22 & 12 \\\hline 25.25 & 15.25 \\\hline 45.125 & 35.125 \\\hline\end{array}$$
Use the data in each table to find an equation that mathematically describes the relationship between the two quantities.$$\begin{array}{|c|c|}\hline \text { Seasonal employees } & \text { Employees } \\\hline 25 & 75 \\\hline 50 & 100 \\\hline 60 & 110 \\\hline 80 & 130 \\\hline\end{array}$$
Explain the difference between an expression and an equation. Give examples.
Use each word below in a sentence that indicates a mathematical operation. If you are unsure of the meaning of a word, look it up in a dictionary.$$\begin{array}{llll}\text { quadrupled } & \text { deleted } & \text { bisected } & \text { garnished } \\ \text { confiscated } & \text { annexed } & \text { docked } & \text { quintupled }\end{array}$$
Use each of the variables $a, b, c,$ and $d$ only once to write:a. a sum of two differencesb. a difference of two sums
Fill in the blank: If $T=16 s$ and $s=\frac{r}{2},$ then $T=r$