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.

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

• Seán has 12 € more than Aoife.

• Seán has twice as much as Aoife.

• Seán has 5 € less than three times what Aoife has.

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

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 $a$.

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.

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) Aoife'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.

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.

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.

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 $x = 8$

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

1. a) Identify and explain the two mathematical errors made by Cian 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.

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?

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

To calculate the numerical value of the algebraic expression:

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.

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.

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) $\frac{10x^3y}{2x^3y}$