In this edition of Zimaths we take a closer look at one of the most mysterious of mathematical concepts - none other than Mr x himself. At first it seems that x is the most elusive and abstract concept but it helps to remember that it is just a number. We may not know the value of the number immediately but it is nothing more intimidating than any other number. When you are simplifying an 'algebraic expression' (a set of terms with letters in them standing for unknown numbers) you may even find it helpful to put in a value for x to make it more concrete. Look at these two examples:
2x + 3x Imagine x was equal to 4 : Two 4's plus three 4's would give you five 4's altogether and so 2x + 3x is the same as 5x.
x x 2x Again, if x was equal to 4 : 4 x 2 x 4 could be written as 2 x 42 and so x x 2x is the same as 2x2.
Of course, when you are working with an equation instead of just an expression then x has a certain definite value which you are trying to find. In that case you would apply the method of solving equations to get x by itself in order to work out its value.
Let's test your skills in algebra. Try these questions by yourself first, then mark your work and then make very sure you understand the reasoning behind each one. (Questions are taken from an algebraic diagnostic test, source unknown.)
1) In the expression a + 5, 'a' stands for a) 1 c) nothing b) any number d) none of these 2) 3a + 7a = a) 10 apples c) 10a b) 3a + 7a d) 100 3) When is a + b + c = a + z + c ? a) always c) when b = z b) never d) none of these 4) If I multiply 3 by p the result is a) 3 x p or 3p c) only 3p b) only 3 x p d) none of these 5) Which is larger : 3 x n or n + 3? a) 3 x n c) they are the same b) n + 3 d) it depends what 'n' equals 6) If 2xy = 240, and x = 4, what can you say about the value of y? a) y = 30 c) y = 1 b) y = 0 d) y = 60 7) If 2m = 10, what can you say about 'm'? a) m = 8 c) 'm' stands for 'metres' b) m = 5 d) none of these 8) If a = 7, c = 9, what can you say about 'b'? a) b = 6 c) b = 8 b) b = b d) b = 11 9) 6x + 2y + x = a) 7x + 2y c) 8xy b) 8x2y d) 6x2 + 2y 10) Add 3 onto m + 4 a) 3m + 4 c) m + 7 b) 7m d) none of these
(Answers : 1) b 2)c 3)c 4)a 5)d 6)a 7)b 8)b 9)a 10)c )
We come now to a small section of the syllabus relating to algebra where it is easy to make mistakes:
Changing the Subject of the Formula
The syllabus requires you to be able to "change the subject of a formula and substitute in formulae including those from other subjects (e.g. science)." The 'subject of the formula' is the letter which stands alone on one side of the equation. Once you have seen some worked examples you should be able to master this section easily. If we take actual formulae which relate to science then you will see that algebra is not necessarily completely abstract. In these cases, each letter represents some scientific quantity. The trick is to determine what mathematical processes were done to the letter you are trying to isolate and in what order and then to reverse those processes.
Example 1
Consider the case where we know the starting point of an object, we know its velocity and we know the time for which it has been travelling and we want to work out its finishing position. So we need to make x2 the subject of the formula.
List the mathematical processes done to x2: a) subtract x1 b) Divide by t2 - t1 In reverse then : a) multiply both sides by t2 - t1 : v ( t2 - t1 ) = x2 - x1 b) Add x1 : v ( t2 - t1 ) + x1 = x2 or x2 = v ( t2 - t1 ) + x1
Example 2
From Pythagoras' Triangle : Where h is the hypotenuse and a and b are the sides of the triangle at right angles to each other.
Consider the case where we know the hypotenuse and one other side, a, and we want to find the third side, b.
List the processes done to b: a) squared b) add a2 c) square root the expression.
In reverse then : a) Square both sides : h2 = a2 + b2 b) Subtract a2 : h2 - a2 = b2 c) Square root both sides
Here are some questions for you to try 1) Make n the subject of the formula 100 + nu = 40u
2) Make a the subject of the formula s = ut + 1/2 at2
3) Make m the subject of the formula g ( m + a ) = b
Answers :
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