**JSS 3 Mathematics First Term Week 6**

**Topic: FORMULAE: SUBSTITUTION AND CHANGE OF SUBJECT**

Formulae and substitution

A **formula **is an equation with letters which stands for quantities. For example

C = 2πr

Is the formula which gives the circumference, c, of a circle of radius r.

In science,

I = V/R

Is the formula which shows the relationship between the current I amps, voltage, V volts, and resistance, R ohms, in an electrical circuit. In arithmetic,

I = PRT/100

Is the formula which gives interest, I, gained on a principal, P, invested at R% per annum for T years. Sometimes the same letter can stand for different quantities in different formulae. For example, I stands for current in the science formula and I stands for interest in the arithmetic formula. **Formulae ** is the plural of formula.

**Substitution**

To substitute in a formula means to replace letters by their values. This makes it possible to calculate other values.

Example

A gas at a temperature of 0^{0}C has an absolute temperature of T K, where T = θ + 273.

a. Find the absolute temperature of a gas at a temperature of 68 ^{0}C.

b. If the absolute temperature of a gas is 380 K, find its temperature in ^{0}C.

Solution

a. T = θ + 273

when θ = 68

T = 68 + 273

= 341

The absolute temperature is 341 K.

b. T = θ + 273

when T = 380,

380 = θ +273

Subtract 273 from both sides.

380 – 273 = θ

107 = θ

The temperature of the gas is 107 ^{0}C.

Example

The formula W = VI gives the power, W watts, used by an electrical item when a current of I amps flows through a circuit of V volts.

a. An air conditioner on maximum power needs a current of 25 amps in a 120 volt circuit. Find the power of being used.

b. An electric light bulb is marked 100 watts, 240 volts. Find the current required to light the bulb.

Solution

a. W = VI

when V = 120 and I = 25

W = 120 X 25

= 3 000

The maximum power is 3 000 watts.

b. W = VI

when W = 100 and V = 240,

100 = 240I

Divide both sides by 240.

100/240 = I

I = 10/24 = 5/12

The current required is 5/12 amp.

Example

If y = 5x^{2 }– 1, find

a. the value of y when x = -2

b. the values of x when y = 79.

Solution

a. y = 5x^{2 }– 1

when x = -3

y = 5 X (-3)^{2 }– 1

= 5 X (+9) – 1

= 45 – 1

= 44

b. y = 5x^{2 }– 1

when y = 79

79 = 5x^{2 }-1

Add 1 to both sides.

80 = 5x^{2}

Divide both sides by 5.

16 = x^{2}

Take the square root of both sides

√16 = x

x = +4 or -4

Notice that there are two possible values for x. We can shorten this to x = ±4 where ± is short for ‘+ or –‘.

**Change of subject**

**Formula means**

Relationship between two or more variables

Example y = x + 5 where x and y are variables.

**Subject of a Formula means**

The variable on its own, usually on the left hand side.

Example y is the subject of the formula y = x + 5

Changing The Subject Of A Formula means rearranging the formula so that a different variable is on its own.

Making x the subject of the formula y = x + 5 gives x = y – 5

**Example**

Make x the subject of

y=x+3

We require x to be the subject of the formula. The subject is written on the left, so we switch the sides to get x on the left

Switch sides

x+3=y

We require x by itself on the left hand side. But we have x + 3. The inverse of addition is subtraction

We need to subtract 3 from the left side. But, to keep the equality true, we need to subtract 3 from the right side as well.

So subtract 3 from both sides

Subtract 3 from both sides

x+3−3=y−3

Simplify

x=y−3

**Example**

Makexthe subject of

y=x+3

Switch sides

x+3=y

Subtract 3 from both sides

x+3−3=y−3

Simplify

x=y−3

**Example**

Make x the subject of

y=x+m

Switch sides

x+m=y

Subtract m from both sides

x+m−m=y−m

Simplify

x=y−m

When you do a question yourself it is often helpful to write in these key points before you do the actual algebra. It gets you to think of the logic of the process

**Example**

Make x the subject of

y = x − 5

Switch sides

x−5=y

Add 5 to both sides

x−5+5=y+5

Simplify

x=y+5

**Example**

Make x the subject of

y = x − m

Switch sides

x−m=y

Add m to both sides

x − m + m=y + m

Simplify

x=y + m

**Example**

Make x the subject of

y=8x

Switch sides

8x=y

Divide both sides by 8

8×8=y8

Simplify

x=y8

**Example**

Make x the subject of

y = mx

Switch sides

mx = y

Divide both sides by m

mxm = ym

Simplify

x=ym

**Example**

Makexthe subject of

y=x8

Switch sides

x8=y

Multiply both sides by 8

8×8=8y

Simplify

x=8y

Changing The Subject Of A Formula

**Example**

Makexthe subject of

y=xm

Switch sides

x m=y

Multiply both sides by m

m x m=my

Simplify

x = m y

**Example**

Make x the subject of

y=2x + 5

Switch sides

2x+5=y

Subtract 5 from both sides

2x+5−5=y−5

Simplify

2x=y−5

Divide both sides by 2

2×2=y−52

Simplify

x=y−52

**Example**

Make x the subject of

y=m x + c

Switch sides

mx+c=y

Subtract c from both sides

mx+c−c=y−c

Simplify

mx=y−c

Divide both sides by m

mxm=y−cm

Simplify

x=y−cm

**Example**

Make x the subject of

y=3x−7

Switch sides

3x−7=y

Add 7 to both sides

3x−7+7=y+7

Simplify

3x=y+7

Divide both sides by 3

3×3=y+73

Simplify

x=y+73

**Example**

Make x the subject of

y=mx − c

Switch sides

mx − c = y

Add c to both sides

Mx – c + c = y + c

Simplify

mx =y + c

Divide both sides by m

mxm=y+cm

Simplify

x=y+cm

**Example**

Make x the subject of

y=x2+5

Switch sides

x2+5=y

Fractions are more difficult to work with so to make work easier and errors less likely get rid of the fraction first

Multiply EVERYTHING on both sides by 2

2(x2)+2(5)=2(y)

Simplify

x+10=2y

Subtract 10 from both sides

x+10−10=2y−10

Simplify

x=2y−10

**Example**

Make x the subject of

y = xm + c

Switch sides

xm + c = y

Fractions are more difficult to work with so to make work easier and errors less likely get rid of the fraction first

Multiply EVERYTHING on both sides by m

m(xm) + m(c) = m(y)

Simplify

x + cm = my

Subtract cm from both sides

x + cm – cm = my − cm

Simplify

x = my − cm

**Example**

Make x the subject of

y=x4−7

Switch sides

x4−7=y

Fractions are more difficult to work with so to make work easier and errors less likely get rid of the fraction first

Multiply EVERYTHING on both sides by 4

4(x4) − 4(7) = 4(y)

Simplify

x−28=4y

Add 28 to both sides

x−28+28=4y+28

Simplify

x=4y+28

**Example**

Make x the subject of

y=xm − c

Switch sides

xm – c = y

Fractions are more difficult to work with so to make work easier and errors less likely get rid of the fraction first

Multiply EVERYTHING on both sides by m

m (xm)−m(c)=m(y)

Simplify

x − cm = my

Add cm to both sides

x − cm + cm = my + cm

Simplify

x = my + cm

**Example**

Make x the subject of

y=2×3

Switch sides

2×3=y

Multiply both sides by 3

3(2×3)=3(y)

Simplify

2x=3y

Divide both sides by 2

2×2=3y2

Simplify

x=3y2

**Example**

Make x the subject of

y = axb

Switch sides

axb = y

Multiply both sides by b

b(axb) = b(y)

Simplify

ax=by

Divide both sides by a

axa = bya

Simplify

x = bya