Algebra › Section A · Foundations › Topic A.1
The language of algebra
Arithmetic answers one question at a time; algebra writes a rule once that answers every case at once. This topic teaches the words of that language: variables, expressions, terms, and how English phrases become algebra.
From arithmetic to algebra
Arithmetic computes one case at a time: if a ride costs $30 plus $0.08 for each minute, you can work out the cost of a 10-minute ride, then a 40-minute ride, then another, case by case. Algebra does the job once. Write the rule as \(30 + 0.08t\), where \(t\) is the number of minutes, and every ride's cost can be read off instantly. Algebra is not new arithmetic; it is a language for describing calculation patterns, and like any language it has vocabulary (variable, term, coefficient) and grammar (the order of operations) that this course will use constantly.
Variables, constants, and expressions
A variable is a letter, such as \(x\), \(n\) or \(t\), standing for a number that has not been fixed yet. A constant is a number whose value does not change, like the 30 and the 0.08 in the ride rule. An algebraic expression combines variables and constants with the operations of arithmetic: \(2x^2 - 5x + 1\) is an expression, and so is \(30 + 0.08t\). An expression has no equals sign. It is like a phrase in English: it names a quantity, but it makes no claim about it. The number of minutes a ride lasts is a variable; the flat booking fee is a constant.
Evaluating an expression
To evaluate an expression means to work out its value for a particular value of the variable. Replace the letter with the number, wrapped in brackets, and then follow the order of operations: grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. The brackets matter when the value is negative: substituting \(x = -3\) into \(x^2\) gives \((-3)^2 = 9\), while \(-3^2\) means \(-(3^2) = -9\). Wrapping every substitution in brackets is a habit worth keeping for the whole course.
Terms and coefficients
Expressions are built from terms, the parts joined by addition. The expression \(2x^2 - 5x + 1\) has three terms: \(2x^2\), \(-5x\) and \(1\). Subtraction counts as adding the negative, so the sign in front of a term belongs to that term. In each term, the number multiplying the variable part is the coefficient: 2 in \(2x^2\) and \(-5\) in \(-5x\). A term with no variable, like the 1, is a constant term. Naming the pieces this way feels fussy at first, but the vocabulary pays off the moment you start combining terms and solving equations.
Worked example: evaluating an expression
Evaluate \(2x^2 - 5x + 1\) when \(x = -2\). Substitute in brackets: \(2(-2)^2 - 5(-2) + 1\). Exponents come before multiplying, so \((-2)^2 = 4\) and the expression becomes \(2 \cdot 4 - 5(-2) + 1\). Multiply next: \(8 + 10 + 1\). Add: the value is 19. Checking the arithmetic a second way, the terms were 8, 10 and 1, and indeed \(8 + 10 + 1 = 19\). Notice how the bracket kept the two minus signs under control: \(-5\) times \(-2\) is \(+10\), not \(-10\).
Like terms
Two terms are like terms when they have exactly the same variable parts: the same letters raised to the same powers. The terms \(7x\) and \(-3x\) are like terms, so they combine to \(4x\) by adding the coefficients: \(7 + (-3) = 4\). But \(5x^2\) and \(5x\) are not like terms, because \(x^2\) and \(x\) are different powers; \(5x^2 + 5x\) cannot be squeezed into one term any more than "5 apples and 5 pears" can become 10 of one fruit. Constants are like terms with each other: \(-4 + 9 = 5\). Recognising like terms is the key step in simplifying.
Worked example: simplify with a minus before brackets
Simplify \(3(2x - 5) - (4x - 7)\). First distribute the 3 across \(2x - 5\): \(6x - 15\). The minus sign before the second bracket means subtract every term inside, which is the same as multiplying by \(-1\): \(-4x + 7\). The expression is now \(6x - 15 - 4x + 7\). Combine like terms: \(6x - 4x = 2x\) and \(-15 + 7 = -8\), so the answer is \(2x - 8\). Checking with \(x = 1\): the original gives \(3(-3) - (-3) = -9 + 3 = -6\), and the simplified form gives \(2 - 8 = -6\). Both agree, so the simplification is right.
Translating words into algebra
English phrases map onto operations. Sum or more than signals addition, difference signals subtraction, product signals multiplication, and quotient signals division. The classic trap is the phrase "less than": in ordinary English, "five less than a number" starts from the number and takes five away, so it is \(n - 5\), not \(5 - n\). Similarly "seven more than twice a number" is \(2n + 7\). Read your translation back into English to test it: "\(n - 5\)" says "a number, less five", which matches. This back-translation check catches almost every ordering error.
Expressions versus equations
An expression is a phrase; an equation is a sentence. The equation \(2x + 3 = 11\) makes a claim, and it can be true or false: it is true when \(x = 4\) and false otherwise. Finding that value is called solving, and it is the business of the next section. Inequalities such as \(2x + 3 < 11\) make comparison claims instead. For now the goal is fluency with expressions themselves: reading them, naming their parts, and evaluating and simplifying them, because every equation you will ever solve is built from expressions on each side of the equals sign.
Worked example: a pricing rule as an expression
A bike rental shop charges a $30 booking fee plus $0.08 per minute, so a ride of \(t\) minutes costs \(30 + 0.08t\) dollars. For a 125-minute ride: \(30 + 0.08(125) = 30 + 10 = 40\), so $40. For a 180-minute ride: \(30 + 0.08(180) = 30 + 14.40 = 44.40\), so $44.40. Notice what the expression does that arithmetic alone cannot: it is the shop's whole price list in one line, valid for any ride length. The model assumes the shop charges exactly $0.08 for every whole minute with no other fees; a real contract might have rounding rules, which is exactly the kind of detail a written model forces you to notice.
Common mistakes to avoid
The most common slips are all sign slips. Substituting a negative without brackets turns \(x^2\) at \(x = -3\) into \(-9\) instead of 9. A minus before brackets must flip every term inside, not just the first: \(-(4x - 7)\) is \(-4x + 7\). Combining unlike terms, such as writing \(3 + 4x\) as \(7x\), mixes constants with variable terms that cannot merge. And "less than" reverses the order of subtraction: "five less than a number" is \(n - 5\). If a simplification looks wrong, evaluate both versions at a value you choose; if they disagree, a slip is hiding somewhere.
Key takeaways
A variable stands for an unfixed number and a constant is fixed; expressions combine them with operations but make no claim, unlike equations, which are sentences with an equals sign. Evaluate by substituting in brackets and following the order of operations. Expressions are sums of terms, each with a coefficient and possibly a variable part; only like terms, with identical variable parts, can combine. Simplify by distributing and then combining like terms, flipping every sign after a minus. Translate phrases operation by operation, and check "less than" phrases by reading the algebra back into English.
Flashcards
Flashcard
A letter standing for a number not yet fixed
Test your understanding
Question 1 of 5
Evaluate 3x + 7 when x = -4.
Question 2 of 5
How many terms are in the expression 5x² - 3x + 8?
Question 3 of 5
Simplify 5x - 3 + 4x.
Question 4 of 5
Translate: seven less than twice a number n.
Question 5 of 5
A bike rental costs $30 plus $0.08 per minute. Which expression gives the cost of a ride lasting t minutes?