Writing and Evaluating Algebraic Expressions
Your streaming service charges $\$9$ a month plus $\$2$ for every movie you rent. What do you pay if you rent 5 movies? 12 movies? Instead of redoing the math each month, you can write one expression, $9 + 2m$, that works for any number of movies $m$. That is what algebraic language gives you: a short way to say something once and use it every time.
From Words to Expressions
A variable is a letter that stands for a number we don't know yet, or one that can change. An algebraic expression combines numbers, variables and operations. To translate a phrase, look for the key words:
| Words | Operation |
|---|---|
| sum, plus, more than, increased by | $+$ |
| difference, minus, less than, decreased by | $-$ |
| product, times, twice ($\cdot 2$), triple ($\cdot 3$) | $\cdot$ |
| quotient, divided by, half ($\div 2$), a fifth ($\div 5$) | $\div$ |
| square, cube | $x^2$, $x^3$ |
Translate in pieces, from the inside out.
You will also see two helpful words: the previous number is $x - 1$ and the next number is $x + 1$.
Express the following situations related to saving and social networks in algebraic language: a) The number of followers an influencer* has, $x$, increased by $150$. b)…
Express the following situations related to saving and social networks in algebraic language:
- a) The number of followers an influencer has, $x$, increased by $150$.
- b) The price of a movie ticket, $p$, reduced by $\$2$.
- c) Three times the age of my brother, if his age is $y$.
- d) Half the money I have in my piggy bank, where $d$ is the total amount.
Match each phrase with its corresponding algebraic expression: 1. Twice a number plus five units.…
Match each phrase with its corresponding algebraic expression:
-
Twice a number plus five units. a) $x - 5$
-
A number decreased by five units. b) $2x + 5$
-
A fifth of a number. c) $x^5$
-
A number raised to the fifth power. d) $\frac{x}{5}$
Translate these phrases from common language into algebraic language using a single variable $x$: a) The sum of a number and its half. b) Three times the result of…
Translate these phrases from common language into algebraic language using a single variable $x$:
- a) The sum of a number and its half.
- b) Three times the result of subtracting five from a number.
- c) The square of the previous number.
- d) The difference between twice a number and the next number.
Terms, Coefficients and Like Terms
The terms of an expression are the pieces separated by $+$ or $-$ signs. In $7x^2y - 5x + 3$ there are three terms: $7x^2y$, $-5x$ and $3$. The sign belongs to the term that follows it. A term with no variable, like $3$, is the constant term.
A term made of a number times variables is a monomial. It has three parts:
-
the coefficient: the number multiplying the variables,
-
the variable part: the letters with their exponents,
-
the degree: the sum of the exponents of all the variables.
Remember that a letter with no visible exponent has exponent $1$, and a plain number has degree $0$. A lone $-ab^3$ has coefficient $-1$.
Like terms have exactly the same variable part (same letters with the same exponents). Only the coefficients can differ. Like terms can be combined by adding their coefficients, because they count the same kind of thing.
Copy and complete the following table identifying the components of these monomials: | Monomial | Coefficient | Variable part | Degree | | :--- | :---: | :---: | :---: |…
Copy and complete the following table identifying the components of these monomials:
| Monomial | Coefficient | Variable part | Degree |
|---|---|---|---|
| $7x^2$ | |||
| $-ab^3$ | |||
| $\frac{1}{2}x^3y$ | |||
| $-5$ |
From the following list of monomials, identify and group those that are like terms: $5x^2$, $3y$, $-2x$, $\frac{1}{4}x^2$, $8y$, $x^2$, $-5y$, $10x$
From the following list of monomials, identify and group those that are like terms:
$5x^2$, $3y$, $-2x$, $\frac{1}{4}x^2$, $8y$, $x^2$, $-5y$, $10x$
Analyze the following algebraic expressions and complete the information requested in each case: 1. In the expression $7x^2y - 5x + 3$: How many terms does it have? What…
Analyze the following algebraic expressions and complete the information requested in each case:
-
In the expression $7x^2y - 5x + 3$: How many terms does it have? What is the constant term?
-
Given the monomial $-\frac{2}{3}ab^4$: State its coefficient, its variable part and its total degree.
Evaluating Expressions
To evaluate an expression, replace each variable with its value (substitute) and then follow the order of operations.
Calculate the numerical value of the following expressions for the given values: a) $3x + 10$ for $x = 4$ b) $x^2 - 5$ for $x = 3$ c) $2(a + b)$ for $a = 5$ and $b = 2$…
Calculate the numerical value of the following expressions for the given values:
- a) $3x + 10$ for $x = 4$
- b) $x^2 - 5$ for $x = 3$
- c) $2(a + b)$ for $a = 5$ and $b = 2$
- d) $\frac{12}{x}$ for $x = 6$
Ashley is analyzing the growth of her followers on a social network. She started the month with $x$ followers. During the first two weeks, her number of followers…
Ashley is analyzing the growth of her followers on a social network. She started the month with $x$ followers. During the first two weeks, her number of followers tripled. However, in the second two weeks, she lost $150$ followers due to a cleanup of inactive accounts.
Write the algebraic expression that represents Ashley's final number of followers and calculate how many she has if she started the month with $x = 420$.
Fixed Plus Variable
Many real prices combine a fixed part (it never changes) and a variable part (it depends on how many you use). They all translate the same way:
$$\text{total} = \text{fixed part} + (\text{price per unit}) \cdot (\text{number of units})$$Use the sliders to see it: the fixed part $b$ is where the line starts, and the price per unit $a$ controls how steeply the cost grows.
When a letter stands for the answer itself, we write a formula, such as $C = 4 + 3g$. Then we can plug in numbers to get a total.
In a video game shop, the cost of renting a console is a fixed $\$10$ plus $\$2$ for each hour of use. a) Write the algebraic expression for the total cost for a number…
In a video game shop, the cost of renting a console is a fixed $\$10$ plus $\$2$ for each hour of use.
- a) Write the algebraic expression for the total cost for a number of hours $h$.
- b) How much will Ashley pay if she rents the console for $3$ hours?
On a video game streaming platform, the basic monthly fee is $\$12…
On a video game streaming platform, the basic monthly fee is $\$12$. In addition, every time you want to play a "Premium" title outside the catalog, you must pay an additional $\$3,50$.
Write the algebraic expression that represents the total cost a user will pay per month if they decide to play $x$ "Premium" titles.
A recycling center awards "eco points" according to the material handed in: $5$ points for each pound of glass ($v$). $3$ points for each pound of paper ($p$). $10$…
A recycling center awards "eco points" according to the material handed in:
-
$5$ points for each pound of glass ($v$).
-
$3$ points for each pound of paper ($p$).
-
$10$ fixed points for each visit to the center.
Write the formula for the total points $P$ that a person receives in one visit. Once written, calculate how many points a student will receive who hands in $4$ pounds of glass and $8$ pounds of paper.
Percents and Several Steps
Hard problems combine several steps and sometimes several letters. Two ideas make them easy:
-
A percent is a decimal. Paying with a 15% discount means paying $100\% - 15\% = 85\%$ of the price, so multiply by $0.85$. Adding a 10% tip means paying $110\%$, so multiply by $1.10$. "$40\%$ more than $x$" is $x + 0.4x = 1.4x$.
-
Build the expression in layers: first the total of the parts, then the percent applied to that total.
A courier company charges its customers according to the weight of the packages and the delivery distance. The base fare is $\$B$. To this amount is added: $\$2,5$ for…
A courier company charges its customers according to the weight of the packages and the delivery distance. The base fare is $\$B$. To this amount is added:
-
$\$2,5$ for each pound of weight ($w$).
-
$\$0,15$ for each mile of distance ($d$).
If the customer is "Premium", a $15\%$ discount is applied to the total price of the delivery. Write the algebraic expression that determines the total cost of sending a package for a "Premium" customer as a function of $B$, $w$ and $d$.
A streaming platform offers two types of subscription: Basic Plan: A fixed fee of $\$x$ per month plus $\$0,50$ for each new release movie rented ($p$). Premium Plan: A…
A streaming platform offers two types of subscription:
-
Basic Plan: A fixed fee of $\$x$ per month plus $\$0,50$ for each new release movie rented ($p$).
-
Premium Plan: A fixed fee that costs $\$40\%$ more than the basic one, but allows you to watch an unlimited number of movies at no extra cost.
-
Write the algebraic expression for the monthly cost of the Basic Plan.
-
Write the algebraic expression for the monthly cost of the Premium Plan.
-
Write the expression that represents the difference in cost between the Premium Plan and the Basic Plan in a month where $p$ movies are rented. For what values of $p$ would the Basic Plan be more expensive than the Premium Plan? (State only the condition).
Three siblings, Mark, Ashley, and Ethan, decide to save up to buy a present. We know that: Ashley has saved $\$x$. Mark has saved $\$10$ less than three times what…
Three siblings, Mark, Ashley, and Ethan, decide to save up to buy a present. We know that:
-
Ashley has saved $\$x$.
-
Mark has saved $\$10$ less than three times what Ashley has.
-
Ethan has saved half the sum of what his other two siblings have together.
Express in terms of $x$:
-
The money Mark has.
-
The money Ethan has (simplify the expression).
-
The total money saved by the three siblings.
Consecutive Numbers and Special Phrases
To describe numbers that are related, name one with a letter and write the others using it.
| Description | Expression |
|---|---|
| Any number | $n$ |
| The previous / the next | $n - 1$ / $n + 1$ |
| Three consecutive numbers | $n,\ n+1,\ n+2$ |
| An even number | $2n$ |
| An odd number | $2n + 1$ |
| Half of a number | $\dfrac{n}{2}$ |
| The reciprocal of a number | $\dfrac{1}{n}$ |
Why $2n$ for even and $2n + 1$ for odd? $2n$ is always a multiple of $2$, so it is even. Adding $1$ pushes it to the next odd number.
Squares need care too. "The square of the sum of $a$ and $b$" is $(a + b)^2$, and it is not $a^2 + b^2$. Test it with $a = 2$ and $b = 3$: $(2 + 3)^2 = 25$, but $2^2 + 3^2 = 13$. In fact $(a+b)^2 = a^2 + 2ab + b^2$, and $4 + 12 + 9 = 25$ ✓.
Translate the following properties relating to integers into algebraic language: 1. The square of the sum of any two numbers, $a$ and $b$, is equal to the sum of their…
Translate the following properties relating to integers into algebraic language:
-
The square of the sum of any two numbers, $a$ and $b$, is equal to the sum of their squares plus twice their product.
-
The sum of the squares of three consecutive numbers, where $n$ is the smallest of them.
-
The difference between the cube of a number $x$ and the square of its half.
-
The quotient of the sum of a number and its reciprocal, and three times that number.
Perimeter and Area
In geometry, the sides are often expressions. The perimeter is the sum of all the sides. The area of a rectangle is base $\cdot$ height.
For a square with side $x$ feet: $P = x + x + x + x = 4x$ ft, and $A = x \cdot x = x^2$ ft².
Find the algebraic expression that represents the perimeter of the following square whose side measures $x$ feet:
Find the algebraic expression that represents the perimeter of the following square whose side measures $x$ feet:
Find the algebraic expression that represents the area of the following rectangle, knowing that its base measures $5$ units more than its height ($x$):
Find the algebraic expression that represents the area of the following rectangle, knowing that its base measures $5$ units more than its height ($x$):
Calculate the algebraic expression for the perimeter of the following figure, which represents the plan of a small rectangular play area.
Calculate the algebraic expression for the perimeter of the following figure, which represents the plan of a small rectangular play area.
Volume and Cut-Out Shapes
The volume of a rectangular prism is $V = \ell \cdot w \cdot h$. When the dimensions are expressions, multiply them and use the distributive property. For a cut-out problem, subtract the area you remove from the area you start with. The area of a circle is $\pi r^2$.
Here is a sheet of cardstock with a square of side $x$ cut from each corner. The flaps fold up to make a box with no lid.
A cardboard shipping box has a square base of side $x$ and a height that is $5$ in. greater than the side of the base. Draw a diagram of the rectangular prism and write…
A cardboard shipping box has a square base of side $x$ and a height that is $5$ in. greater than the side of the base.
Draw a diagram of the rectangular prism and write the simplified algebraic expression that represents the volume of the box in terms of $x$.
From a square steel sheet with side $x$ inches, an operator must cut out four equal circles, one in each corner, to manufacture an industrial part. The radius of each…
From a square steel sheet with side $x$ inches, an operator must cut out four equal circles, one in each corner, to manufacture an industrial part. The radius of each circle is $r$ inches.
-
Write the algebraic expression for the area of the sheet that is left over after making the four cuts (use $\pi$ as a symbol).
-
If we know that the radius of the circles is a quarter of the side of the sheet, that is $r = \frac{x}{4}$, simplify the previous expression so that it depends solely on the variable $x$.
We want to design a cardboard box without a lid from a rectangular sheet of cardstock measuring $50$ in. long by $30$ in. wide. To do this, a square of side $x$ is cut…
We want to design a cardboard box without a lid from a rectangular sheet of cardstock measuring $50$ in. long by $30$ in. wide. To do this, a square of side $x$ is cut out from each of the corners and the flaps are folded up.
Write the simplified algebraic expression that represents:
-
The area of the base of the box in terms of $x$.
-
The total volume of the box in terms of $x$.
Patterns and General Terms
A pattern of figures gives a list of numbers: how many dots, sticks or tiles in figure $1$, figure $2$, and so on. The general term $a_n$ is a rule that gives the number for any figure $n$.
To find it, ask how much the count grows from one figure to the next. If it grows by the same amount $d$ each time, then $a_n = d \cdot n + c$, and you find $c$ by testing figure $1$.
Sometimes you add up a growing list. Rows of $1, 2, 3, \dots, n$ dots form a staircase. Two staircases fit together into a rectangle that is $n$ rows by $n + 1$ columns, so one staircase is half of it:
When a total is made of two kinds of things, count each kind with its own expression. For the cost, multiply each count by its price and add: $(\text{posts}) \cdot p + (\text{slats}) \cdot l$.
Observe the following sequence of figures formed by dots and determine the general term $a_n$ that allows you to calculate the number of dots needed for any figure $n$.
Observe the following sequence of figures formed by dots and determine the general term $a_n$ that allows you to calculate the number of dots needed for any figure $n$.
A city council wants to install a wooden fence around a new playground. The fence is built by joining rectangular panels using vertical posts. Each panel needs $3$…
A city council wants to install a wooden fence around a new playground. The fence is built by joining rectangular panels using vertical posts. Each panel needs $3$ horizontal slats. To support the structure, a post is placed at the beginning of the fence and another at the end of each panel.
-
Write an algebraic expression that allows calculating the total number of horizontal slats needed for a fence that has $n$ panels.
-
Write an algebraic expression for the total number of posts needed for a fence of $n$ panels.
-
If each post costs $\$p$ and each horizontal slat costs $\$l$, write the expression for the total cost of the materials for a fence of $n$ panels.
Observe the following sequence of figures formed by small hexagons. The first figure has a central hexagon surrounded by six others. In each step, a new layer of…
Observe the following sequence of figures formed by small hexagons. The first figure has a central hexagon surrounded by six others. In each step, a new layer of hexagons is added around the previous figure.
-
Determine how many hexagons layer number $n$ has (without counting those in the inner layers).
-
Find the algebraic expression that represents the total number of hexagons that make up the complete figure when it has $n$ layers around the central hexagon.
(Hint: The number of hexagons added in each layer follows a progression).
Summary
Back to the start: the streaming plan costs $9 + 2m$ dollars, so 5 movies cost $9 + 2(5) = \$19$ and 12 movies cost $9 + 2(12) = \$33$. One expression, every case.