Algebra - 02

Equations

An equation is a statement that equates two algebraic expressions as equal. Thus, (i) 4 x + 1 = 25 and (ii) x + 3 + 2 x + 20 = 3 x + 23 are equations. Each equation has two parts separated by the equality sign – the left-hand side (LHS) and the right-hand side (RHS).

The symbol whose value is required to be found in any equation is called the unknown quantity or variable. Variables are usually denoted by x, y or z. The process of finding the value of a variable is called solving the equation. The value so found is called the root or solution of the equation.

An identical equation or an identity is an equation in which the LHS and the RHS are equal for any values given to the symbols. Thus, eq. (ii) above is an identity, which is clear after one collects the LHS terms.

An equation of condition, or simply equation in the ordinary sense of the term, is an equation where the LHS and RHS are equal only for particular values of the variable(s). Thus eq. (i) above is an equation of condition which is true only when x = 6.

The art of setting up equations

The language of algebra is equations. According to Isaac Newton, in order to find relationships between quantities, one needs to translate a problem stated in one's language to the language of algebra.

Following is an example of how Newton, in his book Arithmetica Universalis, translated a problem into the language of algebra. The problem involves finding the original capitalCapital is the money and/or property used to start a business. of a merchant: (Perelman, Algebra fun, p. 38)

A merchant has a certain sum of money.

x

During the first year he spent 100 pounds.

x - 100

To the remaining sum he then added one-third of it.

( x − 100 ) + ( x − 100 ) 3 = 4 x − 400 3

During the next year, he again spends 100 pounds,

( 4 x − 400 ) 3 − 100 = 4 x − 700 3

and increased the remaining sum by one-third of it.

( 4 x − 700 ) 3 + [ 1 3 ⋅ ( 4 x − 700 ) 3 ] = 16 x − 2800 9

During the third year he again spent 100 pounds.

( 16 x − 2800 ) 9 − 100 = 16 x − 3700 9

After he added to the remainder one-third of it,

( 16 x − 3700 ) 9 + ( 16 x − 3700 ) 27 = 64 x − 14800 27

his capital was twice the original amount.

64 x − 14800 27 = 2 x

Solving the above equation will give the value of x, the original capital.

Principles and rules for solving an equation

The following principles are useful for solving equations:

1. The same quantity can be added/subtracted to both sides of an equation without changing the equality.

e.g. for the equation 4 x + 2 = 14 , subtracting 5 from both sides, 4 x + 2 − 5 = 14 − 5 ⇔ 4 x − 3 = 9 gives the resulting equation 4 x − 3 = 9 , which is essentially the same as the original.

2. Both sides of an equation can be multiplied/divided by the same non-zero number without changing the equality.

e.g. for the equation 4 x + 2 = 14 , dividing both sides by 4, 4 x + 2 4 = 14 4 ⇔ x + 1 2 = 7 2 gives the resulting equation x + 1 2 = 7 2 , which is essentially the same as the original.

3. Transposition: This is the process whereby any term of an equation may be taken to the other side with its sign changed without affecting the equality.

   e.g. for the equation 4 x + 2 = 14 , transposing 2 to the RHS and changing sign, 4 x = 14 − 2 ⇔ 4 x = 12 gives the resulting equation 4 x = 12 , which is essentially is the same as the original.

4. Any factor of the numerator on one side of the equation can be transferred to the denominator of the other side. Similarly, any factor of the denominator on one side of the equation can be transferred to the numerator of the other side.

   e.g. for the equation 4 x + 2 = 14 , 4 x + 2 = 14 ⇔ 4 x + 2 = 2 × 7 On transferring numerator 2 of RHS to the denominator of LHS: 4 x + 2 2 = 7 ⇔ 2 x + 1 = 7 the resulting equation 2 x + 1 = 7 is essentially the same as the original.

Formulae

A formula is a relation between two or more variables which is written in the form of an equation.

Framing a formula

Framing a formula involves obtaining the relationship between the given quantities.

Frame a formula for the following statement: "The number of diagonals, d, that can be drawn from one vertex of an n sided polygon to all other vertices is equal to the number of sides of the polygon minus 3." (Bansal, Mathematics Class IX, p. 58, Ex. 1)

Make a formula for the statement: "The reciprocal of focal length f of a lens is equal to the sum of reciprocals of the object distance u and the image distance v." (Bansal, Mathematics Class IX, , p. 58, Ex. 2)

Changing the subject of a formula

The subject of a formula is the variable which is expressed in terms of other variables and/or constants. It is the sole term on one side of the formula/equation.

Changing the subject of a formula means rearranging a formula so as to make the required variable come up on one side as the new subject.

Make P the subject of the formula in I = P × R × T 100 .

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