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 00C has an absolute temperature of T K, where T = θ + 273.

  1. Find the absolute temperature of a gas at a temperature of 68 0C.
  2. If the absolute temperature of a gas is 380 K, find its temperature in 0C.

Solution

  1. T = θ + 273

            when θ = 68

                        T = 68 + 273

                            = 341

The absolute temperature is 341 K.

  1. 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 0C.

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.

  1. An air conditioner on maximum power needs a current of 25 amps in a 120 volt circuit. Find the power of being used.
  2. An electric light bulb is marked 100 watts, 240 volts. Find the current required to light the bulb.

Solution

  1. W = VI

when V = 120 and I = 25

          W = 120 X 25

              = 3 000

The maximum power is 3 000 watts.

  1. 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 = 5x2 – 1, find

  1. the value of y when x = -2
  2. the values of x when y = 79.

Solution

  1. y = 5x2 – 1

when x = -3

y = 5 X (-3)2 – 1

= 5 X (+9) – 1

= 45 – 1

= 44

  1. y = 5x2 – 1

when y = 79

 79 = 5x2 -1

Add 1 to both sides.

80 = 5x2

Divide both sides by 5.

16 = x2

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

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