Algebraic Language
A delivery company charges a flat $\text{RD}\$80$ plus $\text{RD}\$25$ for each kilogram. How much does it cost to send a package of $w$ kilograms? The answer, $80 + 25w$, works for any package, without repeating the calculation each time. That is what algebraic language gives you: saying in a few letters something that is always true.
From Words to Letters
An algebraic expression combines numbers and letters with operations. The letter (the variable) represents a quantity that can change. Translating means turning words into operations.
| In words | In algebraic language |
|---|---|
| a number increased by 150 | $x + 150$ |
| a number decreased by 5 | $x - 5$ |
| twice a number | $2x$ |
| three times a number | $3x$ |
| half of a number | $\dfrac{x}{2}$ |
| one fifth of a number | $\dfrac{x}{5}$ |
| the square of a number | $x^2$ |
| the next number after a number | $x + 1$ |
| the number before a number | $x - 1$ |
Notice that "decreased by 5" is $x - 5$: you take 5 away from $x$. And "half" means dividing by 2, not multiplying by 2.
You can also translate the other way: from the expression to the phrase. For example, $x^5$ is read "a number raised to the fifth power."
Express the following situations related to savings and social media in algebraic language: a) The number of followers an influencer has, $x$, increased by $150$. b) The…
Express the following situations related to savings and social media 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 $120$ pesos.
- c) Three times my brother's age, if his age is $y$.
- d) Half of the money I have in my piggy bank, where $d$ is the total amount.
Match each phrase with its corresponding algebraic expression: 1. Double a number plus five units.…
Match each phrase with its corresponding algebraic expression:
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Double a number plus five units. a) $x - 5$
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A number decreased by five units. b) $2x + 5$
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One fifth of a number. c) $x^5$
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A number raised to the fifth power. d) $\frac{x}{5}$
Multi-Step Phrases
Long phrases mix several operations or several letters. Translate in parts, from the inside out, and use parentheses when an operation applies to a whole group.
Consecutive, Even, and Odd Numbers
For numbers related to each other, you name one with a letter and write the others starting from it.
| Description | Expression |
|---|---|
| The next number after $n$ | $n + 1$ |
| Three consecutive numbers | $n,\ n+1,\ n+2$ |
| An even number | $2n$ |
| An odd number | $2n + 1$ |
| The next odd number after $2n+1$ | $2n + 3$ |
A single argument with letters covers infinitely many cases. If $n$ is any integer, $2n$ is even and $2n+1$ is odd. So the sum of any two even numbers $2n + 2m = 2(n+m)$ is always even; with examples like $2+4$ or $4+6$ you would only have checked individual cases.
Translate these word phrases into algebraic expressions using a single variable $x$: a) The sum of a number and its half. b) Three times the result of subtracting five…
Translate these word phrases into algebraic expressions 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 number preceding a given number.
- d) The difference between double a number and the next number.
Translate the following properties regarding 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 regarding integers into algebraic language:
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The square of the sum of any two numbers, $a$ and $b$, is equal to the sum of their squares plus double their product.
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The sum of the squares of three consecutive numbers, $n$ being the smallest of them.
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The difference between the cube of a number $x$ and the square of its half.
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The quotient of the sum of a number and its reciprocal, and three times that number.
Rate Plans: Fixed Part and Variable Part
Many rate plans combine a fixed part (it doesn't change) and a variable part (it depends on the quantity). They are always translated the same way:
$$\text{fixed} + \text{price per unit} \times \text{quantity}$$For the delivery service at the beginning: $80 + 25w$.
When there are discounts or increases, you multiply by a factor. A $15\,\%$ discount leaves $85\,\%$ of the price, that is, $0.85 \times$ price. If the total price is a sum, it goes in parentheses: $0.85(B + 150w + 9d)$. A $40\,\%$ increase is the same as multiplying by $1.4$.
Go back to the beginning: for a $4$-kilogram package, $80 + 25 \times 4 = 180$ pesos.
In a video game store, the cost of renting a console is a fixed fee of $600$ pesos plus $120$ pesos per hour of use. a) Write the algebraic expression for the total cost…
In a video game store, the cost of renting a console is a fixed fee of $600$ pesos plus $120$ pesos per hour of use.
- a) Write the algebraic expression for the total cost for $h$ hours.
- b) How much will Lucía pay if she rents the console for $3$ hours?
On a video game streaming platform, the basic monthly fee is $720$ pesos. In addition, every time you want to play a "Premium" title outside the catalogue, you must pay…
On a video game streaming platform, the basic monthly fee is $720$ pesos. In addition, every time you want to play a "Premium" title outside the catalogue, you must pay an extra $210$ pesos.
Write the algebraic expression that represents the total monthly cost a user will pay if they decide to play $x$ "Premium" titles.
A recycling center awards "eco points" based on the material handed in: $5$ points for each kilogram of glass ($v$). $3$ points for each kilogram of paper ($p$). $10$…
A recycling center awards "eco points" based on the material handed in:
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$5$ points for each kilogram of glass ($v$).
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$3$ points for each kilogram of paper ($p$).
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$10$ fixed points for each visit to the center.
Write the formula for the total points $P$ a person receives in one visit. Once written, calculate how many points a student will receive if they hand in $4$ kg of glass and $8$ kg of paper.
Iratxe is analysing the growth of her followers on a social network. She started the month with $x$ followers. During the first fortnight, her number of followers…
Iratxe is analysing the growth of her followers on a social network. She started the month with $x$ followers. During the first fortnight, her number of followers tripled. However, in the second fortnight, she lost $150$ followers due to a cleanup of inactive accounts.
Write the algebraic expression that represents Iratxe's final number of followers and calculate how many she has if she started the month with $x = 420$.
A streaming platform offers two types of subscriptions: Basic Plan: A fixed fee of $x$ pesos per month plus $30$ pesos for each new release movie rented ($p$). Premium…
A streaming platform offers two types of subscriptions:
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Basic Plan: A fixed fee of $x$ pesos per month plus $30$ pesos for each new release movie rented ($p$).
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Premium Plan: A fixed fee that costs $40\%$ more than the basic plan, but allows unlimited movie streaming at no additional cost.
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Write the algebraic expression for the monthly cost of the Basic Plan.
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Write the algebraic expression for the monthly cost of the Premium Plan.
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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).
Numerical Value
The numerical value of an expression is the result you get when you replace the letters with numbers. You substitute and follow the order of operations: parentheses, exponents, multiplication and division, and finally addition and subtraction.
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 $x = 6$
Monomials: Parts and Like Terms
A monomial is an expression with a number and letters multiplied together, such as $7x^2$ or $-\frac{2}{3}ab^4$. It has three parts:
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Coefficient: the number that multiplies.
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Variable part: the letters with their exponents.
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Degree: the sum of the exponents of the letters.
| Monomial | Coefficient | Variable part | Degree |
|---|---|---|---|
| $7x^2$ | $7$ | $x^2$ | $2$ |
| $-ab^3$ | $-1$ | $ab^3$ | $4$ |
| $5xy$ | $5$ | $xy$ | $2$ |
| $-5$ | $-5$ | (none) | $0$ |
An expression with several addends has several terms. In $7x^2y - 5x + 3$ there are three terms; the one with no letter, the $3$, is the constant term.
Two monomials are like terms if they have the same variable part. Only like terms can be added or subtracted: you add the coefficients and the variable part stays the same.
$$5x^2 + 3x^2 = 8x^2 \qquad 5x^2 + 3x \text{ cannot be simplified}$$Copy and complete the following table by identifying the components of these monomials: | Monomial | Coefficient | Literal part | Degree | | :--- | :---: | :---: | :---:…
Copy and complete the following table by identifying the components of these monomials:
| Monomial | Coefficient | Literal 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:
$5x^2$, $3y$, $-2x$, $\frac{1}{4}x^2$, $8y$, $x^2$, $-5y$, $10x$
Analyse the following algebraic expressions and complete the required information in each case: 1. In the expression $7x^2y - 5x + 3$: How many terms does it have? What…
Analyse the following algebraic expressions and complete the required information in each case:
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In the expression $7x^2y - 5x + 3$: How many terms does it have? What is the constant term?
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Given the monomial $-\frac{2}{3}ab^4$: State its coefficient, its literal part and its total degree.
Perimeters, Areas, and Volumes with Letters
Measurements can be given with letters. You write the usual formula and substitute each side with its expression.
For a volume you multiply three measurements. If a box has a square base of side $x$ and height $x + 5$:
$$V = x \cdot x \cdot (x + 5) = x^2(x + 5) = x^3 + 5x^2$$When you have a product of two parentheses, you multiply each term of the first by each term of the second and then combine like terms. So, $(x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$.
This example uses a different sheet, so you should do the problem from the list yourself. For areas with $\pi$, the area of a circle of radius $r$ is $\pi r^2$. If you remove four circles from a square of side $x$, what remains is $x^2 - 4\pi r^2$. If in addition $r = \frac{x}{4}$, then $4\pi r^2 = 4\pi \cdot \frac{x^2}{16} = \frac{\pi x^2}{4}$.
Find the algebraic expression that represents the perimeter of the following square whose side length is $x$ meters:
Find the algebraic expression that represents the perimeter of the following square whose side length is $x$ meters:
Find the algebraic expression that represents the area of the following rectangle, knowing that its base is $5$ units longer than its height ($x$):
Find the algebraic expression that represents the area of the following rectangle, knowing that its base is $5$ units longer than its height ($x$):
Find the algebraic expression for the perimetre of the following figure, which represents the floor plan of a small rectangular play area.
Find the algebraic expression for the perimetre of the following figure, which represents the floor plan of a small rectangular play area.
A cardboard shipping box has a square base with side length $x$ and a height that is $5$ cm greater than the side length of the base. Draw a sketch of the rectangular…
A cardboard shipping box has a square base with side length $x$ and a height that is $5$ cm greater than the side length of the base.
Draw a sketch of the rectangular prism and write the simplified algebraic expression that represents the volume of the box in terms of $x$.
On a square steel sheet with side length $x$ centimeters, a worker must cut out four identical circles, one at each corner, to manufacture an industrial part. The radius…
On a square steel sheet with side length $x$ centimeters, a worker must cut out four identical circles, one at each corner, to manufacture an industrial part. The radius of each circle is $r$ centimeters.
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Write the algebraic expression for the remaining area of the sheet after making the four cut-outs (use $\pi$ as a symbol).
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Knowing that the radius of the circles is a quarter of the side length of the sheet, that is $r = \frac{x}{4}$, simplify the previous expression so that it depends only on the variable $x$.
A cardboard box without a lid is to be designed from a rectangular sheet of card measuring $50$ cm long by $30$ cm wide. To do this, a square with side length $x$ is cut…
A cardboard box without a lid is to be designed from a rectangular sheet of card measuring $50$ cm long by $30$ cm wide. To do this, a square with side length $x$ is cut out from each of the corners and the flaps are folded up.
Write the simplified algebraic expression that represents:
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The area of the base of the box in terms of $x$.
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The total volume of the box in terms of $x$.
Patterns and the General Term
In a sequence of figures, algebraic language lets you write a formula for figure $n$: the general term $a_n$. To find it, count how much it increases from one figure to the next.
If it doesn't always increase by the same amount, look at how what is added at each step grows. In a figure made of layers, the total is the sum of what each layer adds. For example, for $n$ tables in a row, with one chair at each end and two per table on the sides, there are $2n + 2$ chairs.
Observe the following sequence of patterns made with dots and determine the general term $a_n$ that allows you to calculate the number of dots needed for any pattern $n$.
Observe the following sequence of patterns made with dots and determine the general term $a_n$ that allows you to calculate the number of dots needed for any pattern $n$.
A local municipality wants to install a wooden fence around a new playground. The fence is built by joining rectangular panels using vertical posts. Each panel requires…
A local municipality wants to install a wooden fence around a new playground. The fence is built by joining rectangular panels using vertical posts. Each panel requires $3$ horizontal slats. To support the structure, a post is placed at the beginning of the fence and another at the end of each panel.
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Write an algebraic expression to calculate the total number of horizontal slats needed for a fence with $n$ panels.
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Write an algebraic expression for the total number of posts needed for a fence with $n$ panels.
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If each post costs $p$ pesos and each horizontal slat costs $l$ pesos, write the expression for the total cost of materials for a fence with $n$ panels.
Observe the following sequence of patterns made of small hexagons. The first pattern has a central hexagon surrounded by six others. In each step, a new layer of…
Observe the following sequence of patterns made of small hexagons. The first pattern has a central hexagon surrounded by six others. In each step, a new layer of hexagons is added around the previous pattern.
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Determine how many hexagons layer number $n$ has (without counting those in the inner layers).
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Find the algebraic expression that represents the total number of hexagons that form the complete pattern when it has $n$ layers around the central hexagon.
(Hint: The number of hexagons added in each layer follows a sequence).