Exercises: 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.

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.

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: ___

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.

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

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

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

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

• $3x + 5x - x$

• $2a + 7b - a + 3b$

• $4x^2 + 2x^2 - 5x$

• $m + m + m + n + n$

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

• Adrián has 12 € more than Kenia.

• Adrián has twice as much as Kenia.

• Adrián has 5 € less than three times what Kenia has.

• If they put both their money together and the town hall gives them an extra 20 €, they have 100 € in total.

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 $x = 8$

In a class test, Marcos had to calculate the numerical value of the algebraic expression $8 - 2x$ for $x = -3$. On his sheet, he wrote the following:

$$8 - 2 \cdot -3 = 6 \cdot -3 = -18$$

1. a) Identify and explain the two mathematical errors made by Marcos in his solution.
2. b) Explain why it is essential to write parentheses when substituting a negative number for a variable.
3. c) Correct the procedure and calculate the correct numerical value.

The public bike-sharing service of a Spanish city charges users without an annual pass a fixed unlocking fee of $1,20$ € per trip, plus $0,15$ € for each minute of use.

1. a) Write an algebraic expression to calculate the total cost of a trip taken by a user who pedals for $m$ minutes.
2. b) If you use a bicycle for $18$ minutes to go from your home to the library, calculate the exact amount you will be charged using your expression.
3. c) What does the variable $m$ represent in this real context? Reason whether it would make practical sense to calculate the numerical value of the expression for $m = -4$ or for $m = 25,5$.

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

• 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 $12 + 3(x + 1)$ would mean.

Look at the design of this new skate park area that the local council wants to build. The dimensions are given in metres 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 $x = 4$ metres.

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.

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:

• $a = 5$

• $b = 20$

• $c = 15$

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?

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) $4y^2 - (2y^2 - 5y) + 3y$

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

• a) The price of a cinema 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 $x$ euros and spent a quarter on a book and then $10$ euros on a gift.

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

• a) $(3x^2) \cdot (4x^3)$

• b) $(-15a^4b^2) : (3a^2b)$

• c) $2 \cdot (5xy) \cdot (-x^2)$

• d) $\frac{10x^3y}{2x^3y}$

We want to discover an unknown algebraic expression that meets all the following conditions:

1. It uses a single variable, $x$, and its degree is $2$.

2. It is a binomial formed by a term with $x^2$ and a constant term (a number without a variable).

3. For $x = 0$, its numerical value is $-5$.

4. For $x = 3$, its numerical value is $13$.

1. a) Deduce step by step what the constant term must be using the condition given for $x = 0$.
2. b) Find out what coefficient must multiply $x^2$ so that the requested numerical value is met when $x = 3$, and write the complete algebraic expression.
3. c) Without having to perform long calculations, what will be the numerical value of your expression for $x = -3$? Justify your answer by looking at the sign and the exponent.

According to typical meteorological records in the Iberian Peninsula, as you ascend a mountain, the temperature decreases by approximately $1\text{ }^\circ\text{C}$ for every $150\text{ m}$ of altitude. On a spring day, the temperature $T$ (in degrees Celsius) on the slope of the Sierra de Guadarrama according to the altitude $a$ (in meters) can be modeled using the expression:

$$T = 20 - \frac{a}{150}$$

1. a) In the village located at the base ($a = 600\text{ m}$), the thermometer reads $16\text{ }^\circ\text{C}$. Before making the exact calculation, make a reasoned estimate of whether at the summit of the Peñalara peak ($a = 2400\text{ m}$) the temperature will be above or below $5\text{ }^\circ\text{C}$. Explain what you base your estimate on.
2. b) Check your estimate by calculating the exact numerical value of the temperature for the altitude of the summit ($a = 2400\text{ m}$).
3. c) A hiker claims that at $3300\text{ m}$ the temperature would be $-2\text{ }^\circ\text{C}$. Check if the mathematical formula gives that result and critically assess whether it makes sense to apply this model to that altitude in the Sierra de Guadarrama, knowing that its highest point does not reach $2500\text{ m}$.

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 $n = 15$.

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

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

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

• Work out the numerical value (delivery cost) for a $2$ kg parcel 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.

The floor plan of a new youth recreational centre 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 perimetre of the shape.

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

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

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?

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:

• $A + B - \frac{1}{2}A$

• $(A \cdot C) : (2xy)$

• 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 group of friends wants to hire a padel court for several hours ($h$). The hire price is $12$ euros per hour. In addition, they need to hire rackets for everyone. Racket hire costs $3$ euros each for the whole session. If the group consists of $n$ people:

• Write the algebraic expression for the total hire 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 $a$ euros and each connector costs $b$ euros, 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 ($b = a/4$).

To calculate the numerical value of the algebraic expression:

$$E = 5(2x - 3) - 3(4x - 7) + 4x$$

for the value $x = 15$, two students propose different approaches:

• Laura's strategy: Directly substitute the value $x = 15$ into the expression as it is and perform the operations following the order of operations.

• David's strategy: First simplify the algebraic expression by applying the distributive property and combining like terms; then, substitute $x = 15$ into the resulting expression.

1. a) Solve the exercise step by step following Laura's strategy.
2. b) Solve the exercise step by step following David's strategy.
3. c) Check whether the same numerical result is obtained with both methods and explain why.
4. d) Evaluate both approaches: which one requires simpler arithmetic operations? If you had to calculate the numerical value of that same expression for five different values of $x$ ($x = -2$, $x = 0$, $x = 1$, $x = 10$, and $x = 100$), justify in a reasoned way which strategy you would choose.

In the redevelopment project of a pedestrian promenade, square green areas surrounded by protective tiles along their perimeter are being designed. If a single square planter is installed, $8$ tiles are needed to surround it. If several planters are placed adjacent in a row forming a continuous line, the interior planters share perimeter tiles.

Three urban planning technicians have proposed three different formulas to calculate how many tiles $B$ are needed for a row of $n$ planters:

• Lucía's formula: $B = 2(n + 2) + 2$

• Sara's formula: $B = 8 + 2(n - 1)$

• Nerea's formula: $B = 2n + 6$

1. a) Interpret the geometric meaning of the formulas: explain in a reasoned way how Lucía counted the tiles and how Sara counted them to arrive at their expressions.
2. b) Use algebraic operations (the distributive property and combining like terms) to prove whether the three formulas are equivalent to each other.
3. c) The city council has a pallet with $75$ tiles for an avenue. Devise a strategy based on the previous expressions to calculate the maximum number of complete planters in a row that can be surrounded and exactly how many tiles will be left over.