Algebraic Expressions

In the world of video games, it is very common to translate situations into mathematical language. If we let $x$ be the number of gems you have in a chest, express the following situations in algebraic language:

• Triple the number of gems you have.

• The number of gems you will have if you find 15 more.

• The number of gems you have left after spending half on a skin.

• Double your gems minus 5 that you owe to a friend.

Lucía and her brother Adrián are comparing the money they have saved for the upcoming festival. Lucía has $x$ pesos. Express in algebraic language the money Adrián has in each of these independent cases:

• Adrián has 240 pesos more than Lucía.

• Adrián has twice as much as Lucía.

• Adrián has 100 pesos less than triple what Lucía has.

• If they combine their money and the municipality gives them an extra 400 pesos.

Translate these real-life situations into algebraic language using a single variable ($x$):

• a) The price of a movie ticket ($x$) increased by $10\%$ for booking fees.

• b) Lucía's age, knowing that she is three years younger than twice the age of her younger brother ($x$).

• c) The sum of three consecutive integers.

• d) The savings I have left if I had $x$ pesos and spent a quarter on a book and then $200$ pesos on a gift.

On an online gaming platform, the monthly subscription costs a fixed $240$ pesos, but every time you buy a "skin" for your character, you are charged an additional $60$ pesos.

• a) Write an algebraic expression to calculate the total monthly cost if you buy $x$ skins.

• b) If one month you decide to buy $4$ skins, use the previous expression to calculate how much you will pay in total.

• c) Translate into plain English what the expression $240 + 60(x + 1)$ would mean.

In PE class, the teacher asks us to run around a rectangular training field. We do not know the exact measurements, but we know that the length is $L$ meters and the width is $a$ meters.

• Write the algebraic expression for the perimeter of the field.

• If the teacher tells us that the length is twice the width ($L = 2a$), rewrite the expression for the perimeter using only the letter $a$.

Look at the design of this new skate park area that the local municipality wants to build. The dimensions are given in meters using the variable $x$.

• a) Write a simplified algebraic expression representing the total perimeter of the skate park area.

• b) Calculate the perimeter if the value of the unknown is $x = 4$ meters.

The floor plan of a new youth recreational center has an irregular shape made up of a central rectangle and two identical triangular areas at the sides. The dimensions are given in terms of a variable $x$ representing the basic width of the rooms.

• Write the simplified algebraic expression for the total perimeter of the shape.

• Write the simplified algebraic expression for the total area of the recreational center.

• If, to comply with safety regulations, the total area must be at least $150 \, m^2$, check whether the design is valid when $x = 5 \, m$.

The numerical value is the result of replacing the letter with a number. Imagine that the cost of organising a birthday party is given by the expression $15n + 50$, where $n$ is the number of guests.

Calculate how much the party will cost if you decide to invite:

• $n = 10$ friends.

• $n = 20$ friends.

• $n = 25$ friends.

Calculate the numerical value of the following expressions for the given values. Be very careful with negative signs and remember the order of operations:

• $2x + 7$ for $x = -3$

• $x^2 - 5$ for $x = 4$

• $3(a + b)$ for $a = 2, b = 5$

• $\frac{x}{2} + 10$ for $x = 8$

An influencer is analysing the growth of their followers. They have created a formula to estimate the "impact points" ($I$) of their posts:

$I = 10a + 5b - 2c$

Where:

• $a$ is the number of "likes" (in thousands).

• $b$ is the number of times the post is shared.

• $c$ is the number of negative comments.

Calculate the numerical value of the impact ($I$) for the following data:

• $a = 5$

• $b = 20$

• $c = 15$

Look at the following series of figures made with wooden sticks. We want to know how many sticks we will need to build a row of $n$ squares.

If for 1 square you use 4 sticks, for 2 squares you use 7 sticks and for 3 squares you use 10 sticks:

1. a) How many sticks would you add for each new square?

1. b) Write the formula or algebraic expression that lets you calculate the total number of sticks for $n$ squares.

We are building a series of figures with square tiles following a certain pattern. Look at the first three figures in the series:

• a) If $n$ represents the figure number, write an algebraic expression indicating how many square tiles figure $n$ will have.

• b) How many tiles will figure number $25$ have?

• c) Which figure number will have exactly $40$ tiles?

In a nature reserve, the growth of a fern species that expands following a geometric pattern is being studied. In the first week (week $n=1$), the fern has one central leaf. In the second week, three new leaves grow around it. In the third, another five grow.

Observe the pattern of the stages and complete the following tasks:

• Find an algebraic expression for the total number of leaves on the fern in week $n$.

• If a fern has 121 leaves, which week is it in?

• Work out the numerical value for week $n = 15$.

Identifying the parts of a monomial is essential to "speak" algebra. Complete the information for the following monomials mentally or in your notebook:

• $5x^3 \rightarrow$ Coefficient: ___ , Literal part: ___ , Degree: ___

• $-ab^2 \rightarrow$ Coefficient: ___ , Literal part: ___ , Degree: ___

• $\frac{1}{2}y \rightarrow$ Coefficient: ___ , Literal part: ___ , Degree: ___

• $-7 \rightarrow$ Coefficient: ___ , Literal part: ___ , Degree: ___

To simplify expressions, we can only add or subtract like monomials (those with the same literal part). Fully simplify the following expressions:

• $3x + 5x - x$

• $2a + 7b - a + 3b$

• $4x^2 + 2x^2 - 5x$

• $m + m + m + n + n$

In the study of monomials, it is essential to identify their parts in order to work with them. Complete the information for the following list of monomials:

• Monomial A: $-5ax^3$

• Monomial B: $\frac{2}{3}y^2z$

• Monomial C: $-x^2y^2$

• Monomial D: $12$

For each one, state:

1. The coefficient.

2. The literal part.

3. The degree of the monomial.

4. Write a monomial that is like Monomial A.

Simplify the following algebraic expressions by collecting like terms. Remember to apply the sign rules correctly:

• a) $5x^2 - 3x + 2x^2 + x - 8$

• b) $3ab + 5a - 2ab + 7b - 4a$

• c) $-(2x + 5) + 3(x - 1)$

• d) $4y^2 - (2y^2 - 5y) + 3y$

Carry out the following operations with monomials and simplify the result wherever possible. If the result is a number, state it:

• a) $(3x^2) \times (4x^3)$

• b) $(-15a^4b^2) : (3a^2b)$

• c) $2 \times (5xy) \times (-x^2)$

• d) $\frac{10x^3y}{2x^3y}$

Consider the following three monomials: $A = 12x^2y$, $B = -4x^2y$ and $C = 5xy^2$.

Carry out the following combined operations and simplify the result fully:

• $A + B - \frac{1}{2}A$

• $(A \times C) : (2xy)$

• Explain clearly why you cannot add the monomial $A$ to the monomial $C$. What condition would need to change in their algebraic part for it to be possible?

A courier company uses drones to deliver packages. The total cost of a delivery in pesos depends on three factors: a fixed maintenance cost ($F$), the weight of the package in kg ($p$), and the distance traveled in km ($d$).

The formula they apply is: $C = 1.5F + \frac{p \times d}{10} - 0.5p$.

• If the fixed cost is $F = 4$ pesos, simplify the expression by combining like terms.

• Calculate the numerical value (delivery cost) for a $2$ kg package that needs to travel $35$ km.

• Which factor influences the final price more if increased by one unit: the weight or the distance? Justify your answer based on the structure of the expression.

In a role-playing video game, the damage dealt by a fire spell is calculated using the expression: $D = \frac{3(x + 2y)^2 - 6xy}{2}$, where $x$ is the character's magic level and $y$ is the power of the staff.

• Calculate the damage if the character has magic level $x = 4$ and staff power $y = 1$.

• A programmer claims that if the character has no staff ($y = 0$), the damage simplifies to an expression proportional to the square of the magic level. Show whether this is true by substituting and simplifying.

• If we double the magic level ($2x$) and keep the staff at $y = 0$, by how many times does the total damage increase?

A group of friends wants to rent a padel court for several hours ($h$). The rental price is $240$ pesos per hour. In addition, they need to rent rackets for everyone. Racket rental costs $60$ pesos each for the whole session. If the group consists of $n$ people:

• Write the algebraic expression for the total rental cost ($C$) in terms of the number of hours ($h$) and the number of people ($n$).

• If they decide to split the total cost equally, write the expression for the amount each person must pay.

• Simplify the previous expression. Does the portion corresponding to the rackets depend on the number of people attending? Justify your answer.

Marta is designing modular shelving. Each square module requires $4$ long boards and $4$ corner connectors. However, when joining modules side by side, they share a vertical board and the two connectors at its ends.

• Write an algebraic expression that gives us the number of planks needed to assemble a row of $x$ modules.

• If each plank costs $a$ pesos and each connector costs $b$ pesos, write the expression for the total cost of a row of $x$ modules.

• Simplify the expression for the total cost if we know that the connector costs a quarter of a plank ($b = a/4$).