Algebra Toolbox Part 3

8/13/2019 Algebra Toolbox Part 3

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The AlgebraToolbox

Part Three

Subject Changing(akaTransposition)

Simultaneous Equations


2/21

The Algebra Toolbox

Contents

Section Topics Page1. Changing the Subject a. Introduction 3

b. Examples 4

2. Simultaneous Equations a. Introduction 12

b. Substitution method 13

c. Elimination method 16


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1. CHANGING the SUBJECT

Often we need to rewrite an equation in a different format.

For example, the equation

x + y = 4

may need to be rewritten so that there is one letter on its own on the left.

This means we have to cut n paste!

Lets get y alone on the left, by cut n pasting the x to the right.

Before we do, remember the equation is really

+x + y = 4Now cut n paste, remembering to switch the + x to x

y = 4 x

We could have put the x in front of the 4, and written it as

y = x + 4

So eithery = 4 x, or y = x + 4 would have done.

This is called Making y the subject

In the example above,

x + y = 4

we might have needed to get x = rather than y =. The method is still the

same!

This time cut n paste the y to the other side:

x = 4 y or x = y + 4 is the answer!!

This is called Making x the subject

The Algebra Toolbox 2003 R. Bowman. All rights reserved.3


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Example 1

Make b the subject of

a + b = c

Realising we want b alone, we need to get rid of the a.Cut n paste a to the right:

+a + b = c becomes

b = c a ANSWER!!

Example 2

Make y the subject of

a + x y = 6

METHOD 1

Cutn paste the y to the right, and the6 to the left:

a + x y = 6

a + x 6 = ySwap left & right sides (no signchanges)

y = a + x 6ANSWER !!

METHOD 2

Cut n paste the a and x to theright:

a + x y = 6

y = 6 a x

Now we must have just y, we canthave y so we need to change allthe signs:

y = 6 a x becomes

+y = 6 + a + x

y = 6 + a + xANS!!


Can you see what was behindMethod 1? Getting the y across tothe right got rid of the minus sign in

front, and made it just y. This is asmart move! Minuses can be a pain!


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Example 3

Make p the subject of

2p + y = 3x

To get p alone, there are two things we have to get rid of.the y and the 2.

Remember we can get rid of the y by cut n paste (because its connected by a + sign)

2 by dividing! (because its connected by a sign)

Always do cut n paste first:

2p + y= 3x becomes

2p = 3x y

Now get rid of the 2 by dividing all three terms by 2:

22

3

2

2 yxp=

The reason for doing this division by 2 is so the 2s will cancel and leaveyou with just p !!

22

3

2

2 yxp=

22

3 yxp = ANSWER!!

OR.you could also have written it as2

3 yxp

= !!



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Example 4

Make a the subject of

2ab y =2

3x

Get rid of any fractions first before you do anything else. Remember what todo?

There are 3 terms. Theres a 2 in the bottom of the fraction.Multiply everything through by 2 !!

Remember

3x2 = 3x !!

2

4ab 2y = 3x

Now we have to get rid of the:

2y (use cut n paste because this is connected by a sign)

the 4 and the b (use division because these are connected by a sign)

Cut n pasting the 2y to the right:

4ab2y = 3xbecomes:4ab = 3x +2y

Dividing by 4b

a =b

yx

4

23 +.ANSWER

Could also have writtenb

y

b

xa

4

2

4

3+= which equals

b

y

b

x

24

3+



7/21

Example 5

Make t the subject of

yx

ptw2

5=

Can you remember the first step? If you said Get rid of the fractionfirst youdbe right!!

There are 2 terms, and an x is in the bottom. Multiply both sides by x.

x

ptw 5x = 2y x

Cancelling the xs on the left

5wpt = 2xy

Remember were interested in the t. It has a minus in front. Do you rememberour smart trickfrom before, where we move it across to the right to get rid of theminus?

5w 2xy =pt

All that now remains is thepto get rid of. Remember because thepis connectedto the tby multiplication, we must divide.

Divide all 3 terms byp:

tp

xyw=

25.ANSWER!



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Example 6

Make uthe subject of

yu

a=

First step? Yep. Get rid of the fractionfirst !!

There are 2 terms, and a u is in the bottom. Multiply both sides by u.

u

a u = y u

Cancelling the us on the left,

a = yu

Remembering we want ualone, we divide both sides by y, because yisconnected to uby multiplication.

y

a =

y

yu

Cancel the ys on the right:

y

a =

y

yu

ya = u.ANSWER!!!



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Example 7

Make pthe subject of

t 3p=x(4 +yp)

First step is to get rid of the brackets!!

t 3p=x4 + x yp

t 3p = 4x + pxy

Now we cut n paste trying to get thep terms together on one side.

Lets move the ps to the right(Why??)and the non-ps (t and 4x) to the left.

t 3p = 4x + pxy

t 4x= pxy + 3p

Once theps are together on the right, we factorise. This brings thepout thefront, and means we now have only oneppresent!

t 4x=p(xy+ 3)

We now need to get rid of the (xy + 3). This is connected topby multiplication,so what do we do???

DIVIDE both sides by (xy + 3) !!

)3(

)3(

)3(

4

+

+=

+

xy

xyp

xy

xt

Cancel the (xy + 3) on the right!

pxy

xt=

+

)3(

4.ANSWER!!

BEWARE!!! The term wewant to isolate, thep,APPEARS TWICE!! In youranswer, you can only havethepappearing once, so

special care is needed!!

NoteRight back at the start, had youtaken the p terms to the left rather thanthe right side, you would have ended upwith

3p pxy = 4x tp( 3 xy) = 4x t

p =xy

tx

3

4

THIS IS EXACTLY THE SAME

AS THE ANSWER AT LEFT



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Example 8

Make y the subject

pway

xy23

4=

First get rid of the fraction.Multiply both sides by (3 ay)

ay

xy

3

4 (3 ay) = 2pw (3 ay)

Cancel the (3 ay) on the left

Expand the right to get rid of the brackets

4xy = 6pw 2apwy

Again we have the y occurring twice.Remember what to do?Move the ys to one side. LEFT side is best this time. (means the ys wont have minuses)

4xy+ 2apwy= 6pw

Factorise the left.Remember this means the y will only appear once (we want this to happen)

y(4x+ 2apw) = 6pw

Divide both sides by (4x + 2apw)

)24(

6

)24(

)24(

apwx

pw

apwx

apwxy

+

=

+

+

Now cancel the (4x + 2apw) on the left:

)24(

6

)24(

)24(

apwx

pw

apwx

apwxy

+

=

+

+

)24(

6

apwx

pwy

+

= ANSWER!!!



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Practice Exercises 1

Make the bracketed letter the subject

1. a b = c [a] 9. 4=

a

yxz [x]

2. 2w + c = d [w] 10. a b = 2x [a], [b]

3. 3a + bd = y [d] 11. w kw = x + pw [w]

4. 2x + y = ax + c [x] 12. ktxb

a=

[x]

5. 6= ba

x [x]

6. cta

x= [x]

7. va

x=

32 [x]

8. pv

ux = [v]

ANSWERS



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2. SIMULTANEOUS EQUATIONS

(a) INTRODUCTION

Sometimes you will have a pair of equations containing two variables (like

a and b), and be asked to find the values of the letters that make the

equations true. For example, the equations

a + b = 7

2b a = 5

are bothsatisfied when a = 3 and b = 4.

In fact, a = 3 and b = 4 are the only answers that fit boththese equations!

You will note that a = 5, b = 2 certainly fits the first equation, but when we feed these numbers

into the second equation, it becomes 22 5 = 5, which of course is not true! So, a = 5, b = 2 isno good (not a solution) because it fits only oneof the equations. You can find heaps of numberslike a = 5, b = 2 which will fit only one of the equations, but theyre useless! Were only interestedin the numbers that fit both equations, and they are of course, a = 3 and b = 4.

So, we say that a = 3 and b = 4 are the solutions to the

simultaneous equations

a + b = 7; 2b a = 5.

There are two good methods to solving simultaneous equations. The first is

called substitution. The second is called elimination.



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(b) SOLVING SIMULTANEOUS EQUATIONS BY

SUBSTITUTION

This is best learned by examples.

Example 1

Solve

a + b = 7 Equation (1)

2b a = 5 Equation (2)

(These are the ones from the previous page!)

Method:

Find a lone letter a letter without a number multiplying it. There are 3of these here (the a and b in the 1

stequation, and the a in the 2

nd

equation).

Make one of these the subjectof its equation.Lets pick the a in the top equation:

a + b = 7Use cutnpaste to get a on its own:

a = 7 bNow substitute this into the 2

ndequation:

2b a = 5 now becomes

2b (7 b) = 5

This is now an equation with only one letter!We can solve it!! (See Toolbox 1if youve forgotten!)

2b (7 b) = 5

2b 7 +b = 5

3b 7 = 5

3b = 12b = 4

We have b. Now we need to find a.

We know b is 4, so replace b with 4 in either Equation (1) or (2).

Equation 1 (the simpler!) becomes a + 4 = 7, and so

a = 3 Final answer is a = 3, b = 4 !!

Remember thesign change??

When substituting somethingwith 2 terms like 7 b, itssmart to put it in brackets! Youlsee why belo

lw.



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Example 2

Solve

2x 3y = 19 Equation (1)

5x + y = 22 Equation (2)

Method:Find the lone letter a letter without a number multiplying it. There isonly oneof these here (the y in the 2

ndequation).

Make y the subjectof Equation 2.

5x + y = 22Use cutnpaste to get y on its own:

y = 22 5xNow substitute this into the 1st equation:

2x 3y = 19 now becomes

2x 3(22 5x) = 19

Now we have only one letter!

Why do you think we put the22 5x in brackets? Because ifwe dont, we could overlook thesign change (below in blue).

2x 3(22 5x) = 19

2x 66 +15x = 19Remember thesign change??

17x 66 = 1917x = 85

x = 5

We have x. Now we need to find y.

We know x is 5, so replace x with 5 in either Equation (1) or (2).

Equation 2 (the simpler!) becomes 5 5 + y = 22

25 + y = 22

y = 3

Final answer is x = 5, y = 3 !!



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Example 3

Solve

3a + 2b = 15 Equation (1)


Method:This time theres no lone letter so go for the letter with the smallestcoefficient (multiplying number in front). The b in the 1

stequation is the

best bet.

Make b the subjectof Equation 1.

3a + 2b = 15Use cutnpaste to get rid of the 3a:

2b = 15 3aNow divide through by 2

b =2

3

2

15 a

Now substitute this into Equation 2:

4(2

3

2

15 a )

= 4 2

15 4

2

3a

= 2 15 2 3a= 30 6a

5a + 4b = 26 now becomes

5a + 4(2

3

2

15 a ) = 26

Expand the brackets & cancel the fractions

5a + 30 6a = 26

30 a = 26

a = 26 30

a = 4

a = 4

a = 4

We have a. Now we need to find b.

We know a is 4, so replace a with 4 in either Equation (1) or (2).Equation 1 becomes 3 4 + 2b = 15

12 + 2b = 15

2b = 15 12

2b = 3

b = 1

Final answer is a = 4, b = 1 !!



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(c) SOLVING SIMULTANEOUS EQUATIONS BY

ELIMINATION

This method is an alternative to substitution. It is a good method to use when

there is no letter on its own, e.g. like in the previous Example3, where all

4 letters had numbers attached to them.

Example 4

Solve (by elimination)


5a 4b = 14 Equation (2)

The trick here is to make the numbers in front of one of the lettersmatch, and have opposite signs. Watch what happens:

We are going to double Equation 1 right through:

Equation (3)6a + 4b = 30

5a 4b = 14 Equation (2)

What do you notice about these 2 equations?

The key: the numbers in front of b are now the same, withopposite signs. This is what we want to try to aim for!

Once we have done this, we now work with the new equations, Equation (3) andEquation (2).

Add Equation (3) to Equation (2), adding like terms vertically:

11a + 0 = 44Getting zero meansweve eliminatedb!!11a = 44

a = 4

Now feed a = 4 into any of the Equations (1), (2) or (3) to find b. (Ill choose Equation (1)because the b has a 2 in front of it (smaller is easier!)

3a + 2b = 15

3 4 + 2b = 15

12 + 2b = 15

2b = 3

b = 1 ..Final answer a = 4 and b = 1



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Example 5

Solve (by elimination)



We want make the numbers in front of one of the letters match, andhave opposite signs. Lets choose the a terms, because theirnumbers are smaller overall than those in front of the bs.

2a + 3b = 9

3a + 4b = 11

We can turn a 2 and a 3 both into a 6 easily (2 and 3 can be multiplied to make 6)

The as have a 2 and a 3in front. How can I makethem the same number?

2a + 3b = 9 Multiply this through by 3 6a+ 9b = 27

3a + 4b = 11 Multiply this through by 2 6a + 8b = 22

What do you notice about these 2 equations?

The numbers in front of a are now the same, but we need oppositesigns (so the numbers will cancel when we add). So change all the

signs in one of the equations (doesnt matter which one).

9b + 8b = 9b 8b = 1b

27 + 22 = 27 22 = +5

6a+ 9b = 27

6a 8b = 22

Add like terms (vertically)

0 + b = 5

b = 5

Now feed b = 5 into Equation (1) up the top:

2a + 3b = 9

2a + 15 = 9

2a = 9 15

2a = 6

a = 3..FINAL ANSWER a = 3 , b = 5



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Example 6

Solve the equations

9 + x = y Equation (1)

x + 2y = 12 Equation (2)

by both methods

Substitution

Make x the subject of Equation (1)

x = y 9

Substitute this into Equation (2)

x + 2y = 12 becomes

(y 9) + 2y = 12

Tidy up by collecting y terms

3y 9 = 12

3y = 21

y = 7

Substitute y = 7 into Equation 1 (or 2)

9 + x = y becomes

9 + x = 7

x = 7 9

x = 2

Final answer x = 2, y = 7

Elimination

Rearrangethe equations so the xs, ysand numbers (the 9 and the 12) are alignedvertically:

x y = 9

x + 2y = 12

(Did you follow how the top equation cameabout? the 9 and the y were cut n pasted)

Now noticing the xs both already match(number in front of each is 1), all we haveto do is a sign swap on one of theequations. Lets change all signs in Eq (1):

x + y = 9

x + 2y = 12

Now add:

0 + 3y = 21

3y = 21

y = 7

Feed this into either equation up top

9 + x = y

9 + x = 7

x = 7 9x = 2

Final answer x = 2, y = 7



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So how do I know which method to use?

The general rule is:

If there is a letter on its own, then its usually easiest to make that

letter the subject and use substitution, as in Examples 1, 2 and 3.

If theres no letter on its own, then elimination is usually the best

method, as in Examples 4 and 5.

Sometimes youll have a choice, in which case its best to use the

method youre most comfortable with!!

Practice Exercises 2

Solve these pairs of simultaneous equations. Use either method.

1 x + y = 8

x y = 2

9 x + y = 6

x y = 1

2 x y = 2

2y = x + 1

10 3x 5y = 13

2x + 5y = 8

3 3 = 2x + y

4x + 6 = 10y

11 y = x 2

x = 2y 1

4 x y = 2

3x + y = 10

12 y = 6 x

2x + y = 85 2x y = 6

3x + y = 1413 10x + 3y = 12

3x + 5y = 20

6 x + y = 5

4x + y = 1414 3x + 4y = 15

3x + 2y = 12

7 2x + 3y = 23

x + 3y = 2215 2x + 3y = 8

2x + y = 4

8 c + 2d = 6

9c + 2d = 5416 3a 5b = 3

4a 5b = 1

Remember torearrange!See Ex 6

Remember torearrange!See Ex 6 Remember to

rearrange!See Ex 6

Remember torearrange!See Ex 6

ANSWERS



20/21

Changing the subject of an equation A quick example

This means that you identify a specific letter you want to get alone on one side

of the = sign. You then use cut n paste to remove everything else from

that side, leaving the letter alone.

e.g. if we wanted to make y the subject of the equation 2y x = 7,

we have to remove the x and the 2 from the left side, which will then leave us

with x as the subject.

2y x = 7

2y = x + 7 cut n paste the x from the left to the right (becomes + x)

y =22

7+

x divide all 3 terms by 2



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ANSWERS Exercise 1

(1) a = b + c (2) w =2

cd (3)

b

ay 3=d (4)

a

ycx

=

2 (5) x = ab 6a (6) x = ac + at

(7)2

3+=

avx (8) px

u

=v (9) z

ya +=

4x (10) a = b+2x; b = a 2x

(11)pk

xw

=

1 (12)

kt

abkx = or

kt

bkax

=

ANSWERS Exercise 2 (in alphabetical order x first, then y)

(1) 5 & 3 (2) 5 & 3 (3) 1 & 1 (4) 3 & 1 (5) 4 & 2 (6) 3 & 2(7) 1 & 7 (8) 6 & 0 (9) 3 & 2 (10) 1 & 2 (11) 5 & 3 (12) 2 & 4

(13) 0 & 4 (14) 3 & 1 (15) 1 & 2 (16) 4 & 3


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Algebra Toolbox Part 3