What’s the difference between an expression and an equation?

Well, I’ll answer that in a moment. More generally, you should be familiar with algebraic terminology since relevant words are used in exam questions—you need to know exactly what they mean. Let’s start at the beginning and build up…

Constants and variables

A constant is a fixed number, such as 2. A variable is a letter which can take on different values, such as x. To complicate things, letters can sometimes be used to stand for constants! For example, I can define a quadratic expression as ax^2+bx+c, where a, b and c represents constants, and x is the variable. Or, as another example, I could say, the line y=3x+c goes through the point (1, 6)—find the value of the constant c. I’m sure you can easily work out that c=3, thus the line has the equation y=3x+3. Now, since x and y are variables, they can take on different values, representing different points on the line: for example, (0, 3), (-2, -3), (2.1, 9.3).

Generally, the letters used for variables come from the end of the alphabet, and constants from the beginning of the alphabet.

The number which multiplies a variable is known as its coefficient.

Terms 

An algebraic term can consist of numbers and variables which can be multiplied or divided, but not added or subtracted. Powers can also be included. Thus, all the following are terms: 3, y, 3y, 3y^2, 4xy, 5x^2y, \dfrac{x}{2}\dfrac{2x^3}{y^2}.

Like terms have the same variables to the same powers: 3x and 5x are like terms, as are 2x^2y and 5x^2y.

The degree or order of a term is the overall power of its variables. Thus, 3y^2 has degree 2, as does 4xywhereas 5x^2y has degree 3.

Expressions

When terms are added or subtracted, we end up with an algebraic expression. Here’s an expression with two terms:  4x+5y.

If an expression contains like terms, it can be simplified: 4x+5y+3x-2y=7x+3y.

Expressions don’t have values unless you assign values to the individual variables. On their own, there’s not a lot you can do with them, except simplify them, or rewrite them by multiplying out or factorising where relevant.

4xy^2+6xy=2xy(2y+3) [or, more strictly, 4xy^2+6xy \equiv 2xy(2y+3), where \equiv means ‘identically the same as’].

So, with an expression, there is nothing to solve. But that’s what you do with…

Equations

An equation is formed when two expressions are made equal, the solution to the equation being the value of the variable which makes the expressions equal.

4x-3=9 and 6x+1=4x+5 are examples of linear equations, where there is only one possible solution. Quadratic equations can have up to two solutions, but beyond this there are no general algebraic methods for equation solving (you can use polynomial factorisation, but most polynomials don’t factorise). To solve an equation such as 2^x=5x+3 requires a numerical method, and this is what your calculator would use to find the solution.

An equation with two variables, such as x+y=8, has an infinite number of solutions: (6, 2), (0, 8), (-4, 12) are just a few possibilities. But if there is a second equation involving the same variables, these can be solved simultaneously resulting in just one solution. Similarly, three equations in three variables have just one solution—as do n equations in n variables.

Formulae

A formula uses algebraic variables to show the connections between actual quantities. Thus, the formula which expresses Pythagoras’ Theorem for the three sides of a right-angled triangle is c^2=a^2+b^2. And the formula connecting the period of swing for a pendulum with its length is T = 2\pi \sqrt {{l \over g}}, where ‘g’ is the acceleration due to gravity (note: this is a constant, but it’s simpler to use the letter ‘g’ than the number which is 9.80665…). So, although a formula looks like an equation, it will only become an equation once you assign values to variables. For example, what length pendulum will have a period of 2 seconds?

We need to solve 2 = 2\pi \sqrt {{l \over g}} for land if you work this through you should find that l = 0.993{\rm{m}}.

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