Algebra - Simple Equations

Equations - An equation is a statement in symbols that two expressions stand for the same number.

In simple words, an equation is an expression of the equality of two numbers.

For example, 3x + 2 = 8 is an equation.

It states that 3x + 2 and 8 stand for the same number. In other words, both are equal in value.

That part of the equation which precedes the sign of equality is called the first member, or left side.

For example, in equation 3x + 2 = 8,
3x + 2 is left side

The part of equation which follows the sign of equality is called the second member, or right side.

For example, in equation 3x + 2 = 8,
8 is right side

We often employ an equation to discover an unknown number from its relation to known numbers (as shown in illustration below)

Illustration-

Sum of ages of Ram and Shyam is 35. If age of Shyam is 18 years, find the age of Ram.

Solution.

Let us assume age of Ram as x years

Ram's age is unknown. But age of Shyam is known. Also, we know the sum of their ages.

From the information given in the question, we can frame an equation as shown below -

x + 18 = 35

Looking at above equation, one can easily  calculate of x.

x will be equal to 17 (Since 17 should be added to 18 to get 35)

So Ram's age is 17 years.

We thus found an unknown number from its relation to known numbers using an equation.

We usually represent the unknown number by one of the last letters of the
alphabets like x, y, z. ( In illustration above, we had assumed age of Ram as x years)

Thus, in the equation ax + b = c, x is supposed to represent an unknown number, and a, b, and c are supposed to represent known numbers.

Simple Equations. An equation which contains the first power of x, the symbol for the unknown number, and no higher power, is called a simple equation, or an equation of the first degree.

Thus, ax + b = c is a simple equation, or an equation of the first degree in x.

Solution of an Equation. To solve an equation is to find the unknown number; that is, the number which, when substituted for its symbol in the given equation, renders the equation an identity. This number is said to satisfy the equation, and is called the root of the equation.

Axioms. In solving an equation, we make use of the following axioms:
Ax. 1. If equal numbers be added to equal numbers, the sums will be equal.
Ax. 2. If equal numbers be subtracted from equal numbers, the remainders will be equal.
Ax. 3. If equal numbers be multiplied by equal numbers, the products will be equal.
Ax. 4. If equal numbers be divided by equal numbers, the quotients will be equal.

If, therefore, the two sides of an equation be increased by, diminished by, multiplied by, or divided by equal numbers, the results will be equal.

Thus, if 8x = 24, then
8x + 4 = 24 + 4,
8x − 4 = 24 − 4,
4 × 8x = 4 × 24,
and 8x ÷ 4 = 24 ÷ 4.

Transposition of Terms. It becomes necessary in solving an equation to bring all the terms that contain the sym- bol for the unknown number to one side of the equation, and all the other terms to the other side. This is called transposing the terms.

We will illustrate by examples:
1. Find the number for which x stands when 14x − 11 = 5x + 70.

The first object to be attained is to get all the terms which contain x on the left side of the equation, and all the other terms
on the right side.

This can be done by first subtracting 5x from both sides (Ax. 2), which gives 9x − 11 = 70,
and then adding 11 to these equals which gives 9x + 11 − 11 = 70 + 11.
Combine, 9x = 81.
Divide by 9, x = 9.

Any term may be transposed from one side of an equation to the other, provided its sign is changed.

Any term, therefore, which occurs on both sides with the same sign may be removed from both without affecting the
equality; and the sign of every term of an equation may be changed without affecting the equality.

Verification. When the root is substituted for its symbol in the given equation, and the equation reduces to an identity, the root is said to be verified.

We will illustrate by
examples:

1. What number added to twice itself gives 24?

Let x stand for the number;
then 2x will stand for twice the number,
and the number added to twice itself will be x + 2x.

But the number added to twice itself is 24.

∴ x + 2x = 24.
Combine x and 2x, 3x = 24.
Divide by 3, the coefficient of x,
x = 8.
Therefore the required number is 8.

Verification. x + 2x = 24,
8 + 2 × 8 = 24,
8 + 16 = 24,
24 = 24.

2. If 4x − 5 stands for 19, for what number does x stand?

We have the equation
4x − 5 = 19.

Transpose −5, 4x = 19 + 5.

Combine, 4x = 24.

Divide by 4, x = 6.

Verification -
4x − 5 = 19,
4 × 6 − 5 = 19,
24 − 5 = 19,
19 = 19.

3. If 3x − 7 stands for the same number as 14 − 4x, what number does x stand for?

We have the equation
3x − 7 = 14 − 4x.

Transpose −4x to the left side, and −7 to the right side,
3x + 4x = 14 + 7.

Combine, 7x = 21.

Divide by 7, x = 3.

Verification -
3x − 7 = 14 − 4x,
3 × 3 − 7 = 14 − 4 × 3,
2 = 2