These notes are for a pupil in Y10, who was not sure how to proceed in a test of basic algebra. This is a first draft, and there will be additions and revisions.
First Notes on Algebra.
Algebra is usually a way of finding out something we don't know. This is usually written by a letter. We are given other information that we have to manipulate to work out the value of this letter.
Algebra does not usually involve big numbers, but we have to make decisions about what we do, and to be careful - often we have to do the same thing twice.
If there's more than one thing we don't know, it becomes complicated.
So, a formula such as y = x + 3 can only give us a range of answers, depending on knowing what either x or y is. If we don't know, we can substitute a number for x and do the calculation. The question will usually tell us what to substiture, so that, if x = 1, y is 1 + 3, if x is 2, y is 2 +3. We can plot the results on a graph, which for this type of formula gives us a straight line.
If we simplify an expression in brackets, we unpack it by carrying out an operation for each item in the bracket. So, (3x⁴)² becomes 9x6. - we have to square both the 3 and the x4
An equation gets its name from the word equal. Usually, equations contain letters and numbers, and we can manipulate the numbers to find a value for the letter. Both sides have to be kept equal, and we do this by doing the same thing to each side.
But how to decide what to do. Look closely at the equation, and then
If there is a fraction in an equation, multiply both sides by the denominator (whatever is below the line). This turns the fraction on one side into a whole, and multiplies the other side to make up for this change.
If both sides contain a number that will divide by a common factor, (eg 2x =4), divide both sides by that factor, so that x =2.
If it contains a negative number, add that number to each side. This will bring the negative number to 0, and keep balance by adding it as a positive to the other side. X - 5 = 0 becomes x =5.
If it contains a positive number, bring that number to zero by subtracting it from each side. X + 5 = 0 becomes x = - 5.