MATHEMATICS SSS ONE

MATHEMATICS SSS ONE

CHANGING THE SUBJECT OF A FORMULA

DAY ONE

Introduction

The ability to rearrange formulas or rewrite them in different ways is an important skill in mathematics. This course will explain how to rearrange some simple formulas.

The subject of a formula

You will be familiar with the formula for the area of a circle which states. A =πr2 Here, A is the area and r is the radius. In the form A =πr2, we say that A is the subject of the formula. Usually the subject of a formula is on its own on the left-hand side. If you know the value of then you can substitute directly to find A. Changing the subject of a formula is exactly the same as solving an equation. The key thing to remember is that ‘whatever you do to one side of the formula you must do to the other side. (Aquinas maths department)

. To rearrange a formula or change the subject of a formula you may;

• add or subtract the same quantity to or from both sides
• •multiply or divide both sides by the same quantity

Example 1

Make x the subject of the formula in each of the following cases.

1. a + x = y + z
2. a + 3x = y + z
3. ax = y+ z

Solution

1. a + x = y + z

a + x - a = y +z – a     (Subtract a from both side)

x = y + z – a

2. a + 3x = y + z

a + 3x = y +z – a       (Subtract a from both side)

3x = y +z – a          (Divide both sides by 3)

X =  y +z – a       3

3. ax =y+ z  (Divide both sides by a)

x =    y +z        a

Exercise 1

Prepare yourself by making x the subject in each of the following cases:

1. y = a + x             Ans (x = y – a)
2. y = 2x – a             Ans (x =y+a2)
3. 2w = 3x             Ans (x= 2w3)
4. ax – y = 2y             Ans (x = 2y+ya)
5. Y = mx + c (m)         Ans ( m = y-cx)

In each case, make the letter at the end the subject of the formula.

1. 2s = 2ut + at2 (a)         Ans ( a =2s-2utt2)
2. Y = mx + c (c)        Ans (c = y – mx)

DAY TWO

Formulae with brackets and fractions

Example 2

Make x the subject of the formula in each of the following cases.

1. a(x + b) = c
2. xa = 1 + yb
3. x+yy = ya + ay

Solution

1. a(x + b) = c

ax + ab = c (Expand the brackets)

ax = c – ab (Subtract ab from each side)

x = c – ab a (divide both side by a)

2. xa = 1 + yb

bx = ab + ay (Remove the fractions by multiplying both sides by ab)

x =   ab+ y b  (Divide both side by b)

3. x+yy =  + ay

a(x + y) = y2 + a2 (multiply both side by ay)

ax + ay = y2 + a2 (expand the bracket)

ax = y2 + a2 – ay (subtract ay from both side)

x = y2 + a2 – ay   (divide both side by a)

Divide both side by a, we have

Exercise 2

In each of the following cases make x the subject:

1. 2 (x + a ) = y             Ans (x = y-2a2)
2. xa = yz                Ans (x = ayz)
3. a(x + y) = y             Ans (x = y ( 1-a)a)

DAY 3

Formulae which require factorizing first.

Example 3

Make x the subject of the formula in each of the following cases.

1. a(x + y) = b(x + y)
2. y = x+ax-a

Solution

1. a(x + y) = b(x + y)

ax + ay = bx + by (Expand both side)

ax – bx = by – ay ( Collect like terms)

x (a - b) = y(b - a) (factorize)

x = y (b-a)a-b (Divide both side by (a – b))

2. y = x+ax-a

y(x - a) = x + a (cross multiplication)

yx – ya = x + a (Expand the bracket)

yx – x = a + ya ( Collect like terms)

x (y - 1) = a(1 - y) (factorize)

x = a (1+y)1-y (Divide both side by (1 – y))

Exercise 3

In each of the following cases make x the subject:

3. x + xy = y          Ans ( x = y1+y)
4. y(x + z) = 3z (x + y)     Ans ( x = 2yzy-3z)

DAY FOUR

Formulae with roots and powers

Example 4

Make x the subject of the formula in each of the following cases.

1. x – 3 = y
2. X2 – y2 = 1

Solution

1. x – 3 = y

x  = y + 3        (First get x on its own)

X = (y + 3)2         (square both sides)

   

2. X2 – y2 = 1

X2 = 1 + y2     (Add y2 to both sides)

X = 1 + y2     (Square both sides)

Exercise 4

In each of the following cases make x the subject:

1. X2 – y2 = a2       Ans ( x=a2+ y2)
2. X3 – y3 = 1     Ans( x = 31+ y3)
3. x – 1 = y     Ans x = (y + 1)2

DAY FIVE

Problems on change the subject of the formulae

Example 5

5ai) Make x the subject of the formula

a + x = y + z

5aii) Hence find x if y = 2, z = 1 and a = 1

5bi) Make x the subject of the formula

a(x + y) = b(x + y)

5bii) Hence find x if y = 2, b = 1 and a = 2

Solution of 5a

5ai) a + x = y + z

a + x - a = y +z – a     (Subtract a from both side)

x = y + z – a

5aii) x = y + z – a

x = 2 + 1 - 1

x = 2

Solution of 5b

5bi)  a(x + y) = b(x + y)

ax + ay = bx + by (Expand both side)

ax – bx = by – ay ( Collect like terms)

x (a - b) = y(b - a) (factorize)

x = y (b-a)a-b (Divide both side by (a – b))

5bii)  x = y (b-a)a-b

x = 2 (1-2)2-1

x = - 2

Exercise 5

1. Make x the subject of the formula in each of the following cases.

x – 3 = y

1. Hence, find x, if y = 3                 Ans (X = (y + 3)2, x = 81)

2. Make x the subject of the formula in each of the following cases.

a + 3x = y + z

2. Hence find y, if x = 1, z = 3 and a = 2        Ans (X =  y +z – a       3 ,  y = 2)