Algebraic Expressions

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

• Three times the 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 a friend.

Ashley and her brother Ethan are comparing the money they have saved for the next festival. Ashley has $x$ dollars. Express in algebraic language the money Ethan has in each of these independent cases:

• Ethan has $\$12$ more than Ashley.

• Ethan has twice as much as Ashley.

• Ethan has $\$5$ less than three times what Ashley has.

• If they pool their money and the city council gives them an extra $\$20.$

On an online video game platform, the monthly subscription costs a fixed $\$12$, but every time you buy a skin for your character, you are charged an additional $\$3$.

• a) Write an algebraic expression that allows you to calculate the total monthly cost if you buy $x$ number of 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 ordinary language what the expression $12 + 3(x + 1)$ would mean.

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

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

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

• c) The sum of three consecutive integers.

• d) The savings I have left if I had $x$ dollars and I spent a quarter on a book and then $10$ dollars on a gift.

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 analyzing 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:

• $a = 5$

• $b = 20$

• $c = 15$

In a role-playing video game, the damage inflicted by a fire spell is calculated with 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 a magic level of $x = 4$ and a staff power of $y = 1$.

• 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. Prove whether this is true by substituting and simplifying.

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

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 terms (those that have the same literal part). Simplify the following expressions as much as possible:

• $3x + 5x - x$

• $2a + 7b - a + 3b$

• $4x^2 + 2x^2 - 5x$

• $m + m + m + n + n$

In the study of monomials, it is essential to identify their parts in order to operate 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, indicate:

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 rule of signs correctly:

• a) $5x^2 - 3x + 2x^2 + x - 8$

• b) $3ab + 5a - 2ab + 7b - 4a$

• c) $-(2x + 5) + 3(x - 1)$

• d) $4y^2 - (2y^2 - 5y) + 3y$

Emma is designing a modular shelving unit. 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 boards needed to assemble a row of $x$ modules.

• If each board costs $\$a$ and each connector costs $\$b$, 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 board ($b = a/4$).

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

• a) $(3x^2) \cdot (4x^3)$

• b) $(-15a^4b^2) : (3a^2b)$

• c) $2 \cdot (5xy) \cdot (-x^2)$

• 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 as much as possible:

• $A + B - \frac{1}{2}A$

• $(A \cdot C) : (2xy)$

• Explain with reasoning why you cannot add the monomial $A$ to the monomial $C$. What condition should change in their variable part to make it possible?

Observe the following series of figures formed with wooden matchsticks. We want to know how many matchsticks we will need to build a row of $n$ squares.

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

• How many matchsticks would you add for each new square?

• Write the formula or algebraic expression that allows you to calculate the total number of matchsticks for $n$ squares.

We are building a series of figures with square tiles following a specific pattern. Observe the first three figures in the series:

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

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

• c) Which figure number will have exactly $41$ 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 carry out the following tasks:

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

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

• Calculate the numerical value for week $n = 15$.

In Physical Education classes, the teacher asks us to run around a rectangular training field. We do not know the exact measurements, but we know that the length measures $L$ feet and the width measures $a$ feet.

• 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 $a$.

Observe the design of this new skate park that the city council wants to build. The dimensions are expressed in feet using the variable $x$.

• a) Write a simplified algebraic expression that represents the total perimeter of the skate park.

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

The layout of a new youth recreation center has an irregular shape composed of a central rectangle and two identical triangular areas on the sides. The dimensions are given in terms of a variable $x$ that represents the basic width of the rooms.

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

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

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

The numerical value is the result of substituting the letter with a number. Imagine that the cost of organizing 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.

A group of friends wants to rent a pickleball court for several hours ($h$). The rental price is $\$12$ per hour. In addition, they need to rent paddles for everyone. The rental of each paddle costs $\$3$ 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 what each person must pay.

• Simplify the previous expression. Does the part corresponding to the paddles depend on the number of people who go? Justify your answer.

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

The formula they apply is: $C = 1,5F + \frac{p \cdot d}{10} - 0,5p$.

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

• Calculate the numerical value (delivery cost) for a $2$ lb package that must travel $35$ mi.

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