EQUATIONS AND TRANSPOSITION OF FORMULAE

1 Introduction to Mathematics for Engineers EQUATIONS & TRANSPOSITION of FORMULAE 1 EQUATIONS AND TRANSPOSITION OF FORMULAE If at any stage you want to return to where you came from use the Explorer Back Button 1. Introduction Most of the Mathematics of Engineering/Science consists of relationships between various physical quantities. These are expressed as Mathematical EQUATIONS . Examples (i) Area of a circle, radius rAr= 2(ii) Coulombs Law. The force of attraction between two charged particles is FkQ Qr=122where QQ12,are the charges, ris their distance apart, kis a physical constant. (iii) Rate of heat transfer Qthrough a slab of heat conducting material, thickness is ()QkT T= 10 where TT10 the is temperature difference, kis the thermal conductivity.

2 EQUATIONS such as the above which represent frequently used results are known as FORMULAE . When using a formula connecting physical quantitites it is of course important to use a consistent set of physical units, but in these notes we are solely interested in the Mathematical rules for manipulating EQUATIONS . 2. EQUATIONS There is only one rule for changing the appearance of an equation. Whatever you do to one side of the equation you must dothe same to the other sideIntroduction to Mathematics for Engineers EQUATIONS & TRANSPOSITION of FORMULAE 2In the following Land Rrepresent whatever may appearon the left and right sides of an equation respectively amd krepresents any numerical or algebraic quantity.

3 If LR=then (i) RL=Example If xyz+=then zxy=+The equation may be written either way. Rather than write 2x=we would write 2x=. Rather than write 2023xx=++we would write 2230xx++=.(ii) LkRk+=+Example If yz x =2then yzz x z += + yx z=+2If you move a negative quantity from one side to the other it becomes positive. (iii) LkRk = Example If yz x+=2then yzz x z+ = yx z= 2If you move a positive quantity from one side to the other it becomes negative. (iv) kLkR=Examples (a) If yzx=+21 Introduction to Mathematics for Engineers EQUATIONS & TRANSPOSITION of FORMULAE 3then zyzzx =+() yzx=+()21(b) If yzx+=12zyzzx+ =12yz zx+=2 This is sometimes called multiplying through by z.

4 (v) LkRk=Example If zxy()+=4then zx yzz()+=4 xyz+=4 This is called dividing through by z.(vii) ()()LRpp=where pis a rational number Examples (i) If xy z+=then ()xyz+=22 xyz+=2(ii) If ()xyz+=2then ()xyz+= xy z+=The following set of examples are concerned with solving EQUATIONS by making use of the above rules. Examples Introduction to Mathematics for Engineers EQUATIONS & TRANSPOSITION of FORMULAE 4(i) x =24(Add 2 to each side) =+=x426(ii) x+=24(Subtract 2 from each side) = =x422(iii) x214+= =x23(Multiply through by 2) = =x236(iv) 4416+=x =412x(Divide through by 2) =+x1243(v) 415x = =46x(Invert both sides) =x416 ==x4623 All the above are examples of LINEAR EQUATIONS .

5 Any equation which can be written in the form 0ax b+=where aand bare real numbers is called a LINEAR EQUATION . The value of xwhich satisfies the equation is called the root of the equation . Linear EQUATIONS have one root bxa= .(vi) x295+=(Square both sides) +=x2925 =x216 (Square root of both sides) = x4(vii) xx22=(Multiply through by x) x222=(Multiply through by 2) x24=x= 2 Introduction to Mathematics for Engineers EQUATIONS & TRANSPOSITION of FORMULAE 5(ix) 34xx+=234xx+=(Multiply through by x ) 2430xx +=(3)(1)0xx =31xorx==(x) 23xx=It is very tempting to divide through by xBUT DON T 230xx =xis a common factor (31)0xx =103xorx== If we had divided through by xthe root at 0x=would have been missed.

6 Examples (vi)-(x) are examples of QUADRATIC EQUATIONS . These are covered in detail in the notes FACTORISATION AND QUADRATIC EQUATIONS . As we saw they may have two real roots , one repeated root , or no real roots . (xi) 3223xx x =It is very tempting to divide through by xBUT DON T 32230xxx =xis a common factor 2(23)0xxx =The expression in brackets factorises (3)(1)0xxx +=03 1xorxor x=== Example (xi) is a CUBIC EQUATION . The general form is 320axbxcx d+++=where ,,abcanddreal numbers . A cubic equation may have three real roots or only one real is a formula for solving cubic EQUATIONS but it is very complicated and not much used instead we tend to use computational methods which will be covered later in your course.

7 Tutorial 1 In exercises 1-10 find the values of xsatisfying the EQUATIONS . 1. x+=652. 279x+= += 4. xx28= ++=1226. xxxx++=+ 12367. 12335xx =+8. xxxx+=+ +1326()() 9. 212x+=10. 1122(1) (1) 0xx +++ =Introduction to Mathematics for Engineers EQUATIONS & TRANSPOSITION of FORMULAE 611. 33122xxx=+ 12. 313 1xx+= +Click here to go the solutions for Tutorial 1. You can use the Explorer back button to return here. 3. TRANSPOSITION of FORMULAE An important application of the above rules above rules occurs in what is called TRANSPOSITION of FORMULAE . We all know the formula for the area of a circle : 2Ar =Hence it is easy to calculate the area of a circle of radius 22( ) == But what if we are told that the area of a circle is 22mand we need to know the radius ?

8 We need to rewrite the formula as Ar =. This is called transposing the formula to make rthe subject of the formula. It is now quite easy to calculate the required radius : == In the following examples the aim is to make the given variable the subject of the formula . 1. 122()kQ QFrr=Rewrite as 212rF kQQ=.Then 212kQ QrF=and finally 12kQ QrF=.2. 101()()kT TQTl =First 10()Qlk TT= then 10 QlTTk= and finally 10 QlTTk=+ 3. 22()abcb=+ First 222abc=+then 222bac= and finally 22bac= Let us look at this in more detail . In the given formula the following operations are carried out on the variable b:- (i) Square it (ii) Add 2c(iii) Take square root To makebthe subject we need to retrace these steps.

9 So we need to carry out the inverse operations in reverse order on the subject of the formula as given which is of (iii) gives 2aInverse of (ii) gives 22ac Inverse of (i) gives 22ac This technique is used in the next example . Introduction to Mathematics for Engineers EQUATIONS & TRANSPOSITION of FORMULAE 73. 224()2 RrSrR +=Operations on R :- (i) Square (ii) Add 2r(iii) Divide by 2 (iv) Square root (v) Multiply by 4r Inverse operations in reverse order (to be performed on S) .(i) Divide by 4r Gives 4Sr (ii) Square Gives 22216Sr (iii) Multiply by 2 Gives 2228Sr (iv) Subtract 2rGives 22228 Srr (v) Square root Gives 22228 Srr Therefore the transposed formula is 22228 SRrr = It needs to be emphasised that the method shown above is of limited application as shown by the next example.

10 4. 12112()RRRRRR=+Before applying the above we would have to rewrite the formula so that the required subject variable only appeared once. In this case it is quicker and easier to proceed as follows: First 1212 RRRRR R+= then 212 112()RRR RRRR RR= = finally 212 RRRRR= 5. qAghAA= 122112(2A)This is a difficult problem . We will do it two ways . First :- Divide both sides by Introduction to Mathematics for Engineers EQUATIONS & TRANSPOSITION of FORMULAE 8 Square both sides qAghAA21222112= Invert both sides AqghAA12221212= Multiply by 2gh21122122ghAqAA= 21122122ghAqAA+= 21222122ghAqqAA+= Invert both sides qghAqAA212222122+=qAghAqA212122222+=And finally AqAghAq211222=+Second way :- Operations on 2A:- (i) Invert.