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ESO Y2 · Mathematics
Algebra · Algebra

Algebraic expressions

A subscription to your favourite music platform costs $x$ euros a month. If you share it with two friends and each pays the same, each share comes to $\dfrac{x}{3}$. You don't know exactly how much the subscription costs, but you already know the calculation that needs to be done. That is algebra.

Before you start, make sure you know: The language of algebra (1st ESO)
Before you start, make sure you know: Operations with integers (1st ESO)

The language of algebra

Diagram illustrating the variable x as a closed box with a hidden value to which three extra lives are added to form the algebraic expression x + 3 x closed box = a number we can't see (starting points in a video game) with x points and 3 extra lives x + 3
Practice★☆☆
Identify the parts of the following monomials by completing the information for each: a) $7x^{2}$ b) $-5ab^{3}$ c) $x^{4}y$ d) $\frac{3}{4}z^{2}$ For each one, state the…

Identify the parts of the following monomials by completing the information for each:

  1. a) $7x^{2}$
  2. b) $-5ab^{3}$
  3. c) $x^{4}y$
  4. d) $\frac{3}{4}z^{2}$

For each one, state the Coefficient, Literal part and Degree.

Solve

Translating phrases into expressions

Each key word becomes an operation. The phrases in 2nd ESO often have several steps: translate from the inside out, as if following a recipe.

Everyday language Algebraic language
Twice a number plus 5 $2x + 5$
Twice (a number plus 5) $2(x + 5)$
Half of a number minus 3 $\dfrac{x}{2} - 3$
A discount coupon of 5 € off the price $x$ $x - 5$
Comparison between the expressions 2x + 5 and 2(x + 5): in the first, the doubling acts only on x, and in the second, on the whole sum "twice a number plus 5" 2x + 5 (only doubles x) "twice (a number plus 5)" 2(x + 5) (doubles the whole sum)
Practice★☆☆
In an online video game shop, the price of a monthly subscription is $x$ euros. Translate the following situations into algebraic expressions: a) The cost of three…

In an online video game shop, the price of a monthly subscription is $x$ euros. Translate the following situations into algebraic expressions:

  1. a) The cost of three monthly subscriptions.
  2. b) The price of a subscription after applying a discount voucher of $5$ euros.
  3. c) What a group of friends pays for a monthly subscription plus a game costing $25$ euros.
  4. d) Double the price of a subscription increased by $10$ euros.
Solve

Numerical value

Calculation of the numerical value of the trinomial x² + 2x + 1 by substituting x with −2 using brackets, giving a final result of 1 x² + 2x + 1substitute x = −2 (−2)² + 2(−2) + 1= 4 − 4 + 1 = 1
Practice★☆☆
Calculate the numerical value of the following algebraic expressions for the given values: a) $5x - 3$ for $x = 4$ b) $x^{2} + 2x + 1$ for $x = -2$ c) $2(a + b)$ for…

Calculate the numerical value of the following algebraic expressions for the given values:

  1. a) $5x - 3$ for $x = 4$
  2. b) $x^{2} + 2x + 1$ for $x = -2$
  3. c) $2(a + b)$ for $a = 3$ and $b = 5$
  4. d) $\frac{10 - x}{2}$ for $x = 6$
Solve

Patterns: the general term

When a figure always grows according to the same rule, you can describe any position $n$ at once with an algebraic expression, the general term $a_n$.

With $n=1$ there are $2$ dots; with $n=2$, $4$; with $n=3$, $6$. Each new position adds $2$ more dots and there is nothing left over at the start: the rule is $a_n = 2n$.

Practice★☆☆
Observe the following sequence of patterns made with dots and determine the algebraic expression for the general term $a_{n}$ (where $n$ is the position of the pattern).…

Observe the following sequence of patterns made with dots and determine the algebraic expression for the general term $a_{n}$ (where $n$ is the position of the pattern). How many dots will the pattern in position $n=50$ have?

n = 1 a₁ = 2 n = 2 a₂ = 4 n = 3 a₃ = 6 n aₙ = ? n = 50 a₅₀ = ?
Solve

Classifying expressions by their terms

The simplest algebraic expressions, with a single term, are called monomials. When they have several terms that are not like terms, they form a polynomial, which is named according to how many terms it has:

Three boxes classifying −7ab² as a monomial, x² + 10 as a binomial and 4x³ − 5x + 2 as a trinomial according to their number of terms −7ab² monomial (1 term) x² + 10 binomial (2 terms) 4x³−5x+2 trinomial (3 terms) with 4 terms or more, simply "polynomial"
Practice★☆☆
Classify the following algebraic expressions, indicating whether they are monomials, binomials or trinomials. In the case of polynomials (binomials and trinomials), also…

Classify the following algebraic expressions, indicating whether they are monomials, binomials or trinomials. In the case of polynomials (binomials and trinomials), also state their degree.

  1. a) $4x^{3} - 5x + 2$
  2. b) $-7ab^{2}$
  3. c) $x^{2} + 10$
  4. d) $3y^{4} - 2y^{2}$
Solve
Practice★☆☆
Simplify the following algebraic expressions by adding or subtracting like terms: a) $3x + 5x - 2x$ b) $7a^{2} - 3a + 2a^{2} + 5a$ c) $4xy + 2x - xy + 3x$ d)…

Simplify the following algebraic expressions by adding or subtracting like terms:

  1. a) $3x + 5x - 2x$
  2. b) $7a^{2} - 3a + 2a^{2} + 5a$
  3. c) $4xy + 2x - xy + 3x$
  4. d) $10n - 5 - 3n + 8$
Solve

Back to the start: If the music subscription costs $15$ € a month ($x = 15$), the numerical value of $\dfrac{x}{3}$ is $\dfrac{15}{3} = 5$ € for each person.

Summary

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