Mathematics · Expressions, equations and inequalities · Grade 7

Expressions: from numbers to letters

A taxi: 5 000 soum plus 2 000 soum per km. What is the fare for x km? How are brackets expanded?

The situation

This is how a taxi app works out the fare: a starting charge of 5 000 soum, then 2 000 soum for every kilometre. The bill for 7 km is 5 000 + 2 000 · 7. Tomorrow you ride 12 km, the day after tomorrow 3 km: instead of writing it afresh each time, we denote the distance by a letter — 5 000 + 2 000x. The length of a garden fence, 2(a + b), is a formula of the same kind. How do we work with expressions with letters: how are brackets expanded, how are like terms collected?

Topic

Numerical and algebraic expressions, formulas, expanding brackets, coefficients, properties of operations. A taxi, an internet tariff and a garden fence: a rule given in words becomes an expression with letters, brackets are expanded, like terms are collected

Key ideas

Algebraic expression
Expanding brackets
Like terms

Formulas

5 000 + 2 000 \cdot x

An algebraic expression: numbers, letters and operations. When a number is put in place of the letter, we get the value of the expression: for $x = 4$ it is 13 000.

P = 2 (a + b), S = a \cdot b

Formulas are equalities with letters: the perimeter and the area of a rectangle for any $a$ , $b$ .

a (b + c) = a b + a c, - (b - c) = - b + c

Expanding brackets: the distributive property; a minus in front of a bracket changes the sign of every term.

2 x + 3 + 4 x - 1 = (2 + 4) x + (3 - 1) = 6 x + 2

Like terms — terms with the same letter part — have their coefficients added.

Theory

A numerical expression and its value

A written form made of numbers, operation signs ( $+$ , $-$ , $\cdot$ , $:$ ) and brackets is a numerical expression: $5 000 + 2 000 \cdot 7$ . The number we get after carrying out the operations is the value of the expression. The order: first what is inside brackets, then multiplication and division, and last addition and subtraction. That is why the taxi bill is $5 000 + 14 000 = 19 000$ , not $7 000 \cdot 7$ .

Algebraic expression

When letters take part, it is an algebraic expression: $5 000 + 2 000 x$ . A letter stands in place of any number; if a particular number is put in, we find the value of the expression for that number: for $x = 4$ , $5 000 + 8 000 = 13 000$ . In the table each $x$ has exactly one value — later this will be called a function. The letter also has permitted values: a distance cannot be negative, and we cannot divide by zero.

Formulas are equalities with letters

If we join two expressions with an equals sign, we get an algebraic equality; an equality that expresses a rule between quantities is called a formula: perimeter $P = 2 (a + b)$ , area $S = a b$ , distance $s = v \cdot t$ . A formula is not for one garden — it works for any $a$ and $b$ ; check it in the third step by dragging the corner.

Expanding brackets and the coefficient

The distributive property $a (b + c) = a b + a c$ : $3 (x + 2)$ is a rectangle of height 3 and width $x + 2$ ; cut in two it gives $3 x + 6$ . If there is a minus in front of a bracket, it is multiplication by $(- 1)$ : $- (x - 4) = - x + 4$ — the sign of every term changes. A numerical factor in front of a letter is a coefficient: in $3 x$ it is 3, in $x$ it is 1, in $- x$ it is $- 1$ .

Properties of operations and like terms

Commutative $a + b = b + a$ , $a b = b a$ ; associative $(a + b) + c = a + (b + c)$ ; distributive $a (b + c) = a b + a c$ . With them we tidy up $2 x + 3 + 4 x - 1$ : we put the $x$ terms side by side, group them, $(2 + 4) x + (3 - 1) = 6 x + 2$ . Terms with the same letter part are like terms; only they can be added: $2 x + 3$ cannot be simplified further.

About the model

The experiment deals only with linear expressions (the letter to the first power) and whole or common-fraction coefficients; the square of a letter and polynomials are in the “Monomials and polynomials” experiment, division by a letter is in “Algebraic fractions”. The taxi tariff is simplified: real apps also add waiting time and a minimum fare. The sliders are limited — but the formulas are true for any number.

Did you know?

The word “algebra” comes from the word “al-jabr” (“restoring”) in Muhammad al-Khwarizmi’s book “Al-kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala”, written in the 9th century.
The equals sign “=” was introduced in 1557 by the Welsh mathematician Robert Recorde, who explained: “no two things can be more equal than a pair of parallel lines”. The consistent use of letters for unknowns was begun by the French mathematician François Viète at the end of the 16th century.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Monomials and polynomials — Grade 7
Special products (algebraic identities) — Grade 7
Algebraic fractions — Grade 7
The linear equation: the balance — Grade 7
Systems of equations — Grade 7
Inequalities: ray and interval — Grade 8
The quadratic equation: the parabola — Grade 8