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DCU School of Mathematical Sciences BASIC SKILLS WORKSHEET 3 Algebra I - Variables and Expressions The aim of this worksheet is to revise some maths that deals with manipulating formulas and expressions. This is the basic ‘grammar’ of maths that allows us to use mathematical language in science, business and engineering. What is algebra? This is the name given to the area of maths that deals with ﬁddling about with symbols and letters. What is it for? It’s for making our lives easier. Up until the 14th Century or so, people used to write things like this: “It is desired that quantity for which thrice the quantity added to twice its own product has the magnitude twelve.” It’s easier to say “Find x where 2x2 + 3x = 12.” What is a variable? A variable is a symbol (usually a letter) that we use to represent a quantity whose value can change or whose value is not known. For example, P = price of a gallon of petrol x = length of a piece of string y = amount of salt in a saline solution It is also common to use letters to represent quantities whose values are ﬁxed and are known. This is very common in science, where examples include c = speed of light in a vacuum G = Newton’s gravitational constant Letters or symbols used in this way are called constants. We also use the word constant to describe √ ﬁxed numbers, e.g. 3, 2, 1 , etc. 2 What is an expression? This means a collection of variables and numbers combined using addition, subtraction, multiplication etc. For example, if x and y represent unknown numbers, then the following are expressions involving x and y: 2x − y 2x + y, xy, 3x2 + 8y, . x + 4y 1 Reminder In order to save time and ink, we use these shorthand ways as substitutes for × (multiplication) and ÷ (division): two quantities written together means multiplication: xy = x × y; two quantities written above and below a horizontal or diagonal bar means division: x = x/y = x ÷ y. y Here are two more deﬁnitions that will be useful to have. Terms A term is an individual part of an expression. For example the expression 2x2 − 4xy + 18y 2 + 21 contains four terms; 2x2 , −4xy, 18y 2 and 21. These are called respectively “the term in x2 ”, “the term in xy” and “the term in y 2 ”. The term which is just a number is called the “constant term”. Coeﬃcients A coeﬃcient is a quantity (usually a number or a constant) that multiplies another quantity. For example, in the expression −2ab + 4c2 + 11bc the coeﬃcient of ab is -2; the coeﬃcient of c2 is 4 and the coeﬃcient of bc is 11. Equations An equation is simply a statement that two expressions are equal to one another. For example, x=4 x2 − 3x + 2 = 0 −2ab + 4c2 + 11bc = 15b E = mc2 The diﬀerence between expressions and equations is that expressions never contain an equals sign. 2 Brackets Suppose that we needed to multiply the expression x − 2 by 3. The result could be written as 3 × x − 2. However we don’t usually use the × sign. Also, we need a way of indicating that all of the expression x − 2 is being multiplied by 3. We use brackets to do this: 3 × x − 2 = 3(x − 2). This means that every term in the brackets gets multiplied by 3, and so the result is 3(x − 2) = 3x − 6. Exercise 1 1. In the following expressions, write down the required quantity. (a) 2x2 − 4x + 3 write down the coeﬃcient of x. 2 (b) 3bc + 5b − abc write down the term in bc. (c) −P 2 + 4P − 32 write down the constant term. 2. A major use of algebra is for writing down rules about e.g. science, economics, engineering in a concise way. For example, consider the total revenue generated by selling a certain number of cars at a certain price. Would could write total revenue = number of cars × price of each car. However it makes life a bit easier if we just abbreviate this by using some deﬁnitions. Let T = total revenue, P = price of car, Q = quantity (or number) of cars. Then we get the simple equation T = P Q. An equation like this which must always be true for the variables T, P, Q is called a formula. In the following examples, invent names (single letters) for the relevant quantities, and write down the formula which is described in words. (a) The average speed of a car is the distance travelled divided by the time taken. (b) The area of a rectangle is the length multiplied by the breadth. (c) Temperature in Farenheit is found by multiplying temperature in Celsius by 1.8 and then adding 32. 3 Substituting in expressions and formulae • This means replacing variables with actual numbers. For example consider the following ex- pressions in u and v: u+v u−v 2u2 + v 2 2u − v . v + 3u If u = 2, v = 3 then these expressions take the values 2+3=5 2 − 3 = −1 2(2)2 + (3)2 = 2(4) + 9 = 8 + 9 = 17 2(2) − 3 1 = . 3 + 3(2) 9 • Making substitutions in formulae allows us to calculate the value of one quantity once the values of all the others are known. For example, the formula s = 4.9t2 gives the distance s (in meters) which an object falls in t seconds when it is dropped. So after t = 1 second, the object has fallen s = 4.9(1)2 = 4.9 metres. After t = 4 seconds, the object has fallen s = 4.9(4)2 = 4.9(16) = 78.4 metres. • Consider the formula f = 3x2 − 4x + 6. We frequently use the notation f (x) = 3x2 − 4x + 6 to emphasise that the formula is telling us how the value of f depends upon the value of x. We might also write T (P, Q) = P Q in the earlier example. If we are using this notation, 4 then f (2) means “the value of the formula when we substitute in x = 2”. So for the example f (x) = 3x2 − 4x + 6 we have f (2) = 3(22 ) − 4(2) + 6 = 12 − 8 + 6 = 10, f (1) = 5, f (−1) = 13, f (−2) = 26. Exercise 2 1. Evaluate the following expressions when x = 3, y = −2, z = 1. (a) x+y (b) 2x + 3yz (c) z−y (d) x2 + y 2 − z 2 2x (e) y + 4z 3xy − 2z 2 (f) 4xyz (g) x2 − 4yz. 2. Find the value of the unknown quantity in the given formula. (a) F = 1.8C + 32 Find F when C = 36. RT (b) P = Find P when R = 4, T = 275 and V = 22. V (c) s = ut + 1 at2 2 Find s when a = 9.8, u = 3.2 and t = 4. 3. Given the formula f (x) = −2x2 + 3x + 6, ﬁnd (a) f (1) (b) f (−1) (c) f (0) (d) f (2) (e) f (−2). Simplifying expressions If you were counting the dots in Figure 1 (p.6), chances are you wouldn’t say that there are 4 dots and 2 dots and 3 dots. You would probably say there are 9 dots. Similarly, in an expression like 2x + 4y 2 + 6x − 3x + 3y 2 it make more sense to put all the same objects together. So the expression is better written as 5x + 7y 2 . When simplifying expressions in this way it is crucial to put only the same objects together. 5 •• • • • • ••• Figure 1: Some dots Multiplying brackets We know that ab means a × b. Similarly, a(2b − c) means a × (2b − c) = 2ab − ac. When two bracketed terms are multiplied, it means that all terms of the ﬁrst bracket are multiplied by all terms of the second. So (x + 2)(x − 3) = x(x − 3) + 2(x − 3) = x2 − 3x + 2x − 6 = x2 − x − 6. and (a − 3)(b + 17) = a(b + 17) + (−3)(b + 17) = a(b + 17) − 3(b + 17) = ab + 17a − 3b − 51. After multiplying out brackets in this way, we always collect like terms together. 6 Exercise 3 1. Simplify the following expressions by collecting together all like terms. (a) −4x + 2y + 6x − 18y (b) 2(−3x + 5y) − 4(x + 3y) (c) 2a2 b + 3a2 b2 − ab2 + 8a2 b. (d) −3(a − b) + 3(−a + b). 2. In each case, multiply out the brackets and simplify the resulting expressions by collecting together like terms. (a) (x + 4)(x + 1) (b) (3y − 4)(y + 1) (c) (2x − y)(3x − 5y) (d) (2a + b − 3)(a + b) Manipulating formulae The formula for converting temperature in celsius (C) to temperature in Farenheit (F ) is F = 1.8C + 32. For example, this allows us to convert 15o Celsius to 59o Farenheit. But what if we wanted to convert from Farenheit to Celsius? Our present formula tells us immediately the value of F in terms of the value of C. We call it an explicit formula for F . We also say that F is the subject of the formula. What we want to do next is make C the subject of the formula, i.e. get C on its own on one side of the formula. We must manipulate the formula to do this. There is only one rule to follow: Whatever we do to one side of the equation, we must also do to the other side. The steps required to do this in the present case are as follows: F = 1.8C + 32 start here F − 32 = 1.8C subtract 32 from both sides F − 32 = C divide both sides by 1.8. 1.8 This completes the process. We now have our Farenheit-to-Celsius formula: F − 32 C= . 1.8 7 Manipulating formulae - examples The rule written down above is the only rule about manipulating formulae. After that, it is a matter of practise. There are some useful guidelines: • If we want to make x the subject of a formula, try to get all the terms involving x together on one side, with anything not involving x on the other side. • If x appears in the denominator of a fraction, multiply both sides of the formula by that denominator. • If there is a square root in the formula, isolate it and then square both sides. 1. Make x the subject of the formula y = −4x − 12. Solution: y = −4x − 12 start here y + 12 = −4x add 12 to both sides y + 12 = x divide both sides by -4 −4 2. Make Q the subject of the formula T = P Q. Solution: T = PQ start here T = Q divide both sides by P P There was only one step here. Dividing by P is just like dividing by any other number. 3. Make a the subject of the formula v = u + at. Solution: v = u + at start here v − u = at subtract u from both sides v−u = a divide both sides by t t 8 4. Make x the subject of the formula 2x + 3 y= x−1 Solution: 2x + 3 y = start here x−1 y(x − 1) = 2x + 3 multiply both sides by x − 1 yx − y = 2x + 3 expand yx − y − 2x = 3 subtract 2x from both sides to get all x’s on the left yx − 2x = 3+y add y to both sides to get everthing not involving x on the right (y − 2)x = 3+y collect terms 3+y x = divide both sides by y − 2 to ﬁnish. y−2 5. Make b the subject of c a=2 . b Solution: c a = 2 start here b c a2 = 4 square both sides b a2 b = 4c multiply both sides by b 4c b = 2 divide both sides by a2 to ﬁnish. a Exercise 4 1. Make Q the subject of P = −3Q + 35. 2. Make t the subject of v = u + at. 2x + 1 3. Make x the subject of y = . x+4 3b − 1 4. Make b the subject of a = . 2b + 5 s 5. Make s the subject of t = 3 . s+1 9 Solutions to exercises Exercise 1 1. (a) -4 (b) 3bc (c) -32. d 2. (a) Let s=average speed, d=distance travelled, t=time taken. Then s = . t (b) Let A=area, x=length, y=breadth. Then A = xy. (c) Let C= temperature in Celsius, F = temperature in Farenheit. Then F = 1.8C + 32. Exercise 2 1. (a) 1 (b) 0 (c) 3 (d) 12 (e) 3 5 (f) 6 √ (g) 17. 2. (a) 96.8 (b) 50 (c) 91.2 3. (a) 7 (b) 1 (c) 6 (d) 4 (e) -8 Exercise 3 1. (a) 2x − 16y (b) −10x − 2y (c) 10a2 b + 3a2 b2 − ab2 (d) −6a + 6b 10 2. (a) x2 + 5x + 4 (b) 3y 2 − y − 4 (c) 6x2 − 13xy + 5y 2 (d) 2a2 + 3ab + b2 − 3a − 3b. Exercise 4 35 − P 1. Q = 3 v−u 2. t = a 1 − 4y 3. x = y−2 1 + 5a 4. b = 3 − 2a t2 5. s = . 9 − t2 11