Linear Equation in two variables: Any first degree equation containing two variables x and y is called a linear equation in two variables. The general form of linear equation in two variables x and y is ax+by+c = 0, where atleast one of a, b is non-zero and a, b, c are real numbers.

Algebra can be thought
of as the next level of study of numbers. If we need to determine anything
subject to certain specific conditions, then we need Algebra. In that sense,
the study of Algebra is considered as “Science of determining unknowns”. During
third century AD(CE) Diophantus of Alexandria wrote a monumental book titled
“Arithmetica” in thirteen volumes of which only six has survived. This book is
the first source where the conditions of the problems are stated as equations
and they are eventually solved. Diophantus realized that for many real life
situation problems, the variables considered are usually positive integers.

The term “Algebra” has
evolved as a misspelling of the word ‘al-jabr’ from one of the important work
titled Al-Kitāb al-mukhtaşar fī hisāb al-jabr wa’l-muqābala (“The Compendious
Book on Calculation by Completion and Balancing”) written by Persian
Mathematician Al-Khwarizmi of 9^{th}
Century AD(CE) Since Al-Khwarizmi’s Al-Jabr book provided the most appropriate
methods of solving equations, he is hailed as “Father of Algebra”.

In the earlier classes,
we had studied several important concepts in Algebra. In this class, we will
continue our journey to understand other important concepts which will be of
much help in solving problems of greater scope. Real understanding of these
ideas will benefit much in learning higher mathematics in future classes.

**Simultaneous Linear
Equations in Two Variables**

Let us recall solving a
pair of linear equations in two variables.

**Linear Equation in two
variables**

Any first degree
equation containing two variables *x* and y is called a linear equation in
two variables. The general form of linear equation in two variables *x*
and *y *is* ax*+*by*+*c *= 0, where atleast one of* a, b *is
non-zero and* a, b, c *are real numbers.

Note that linear
equations are first degree equations in the given variables.

·
*xy – *7 = 3* *is not a linear equation in two variables since the
term* xy *is of degree 2.

·
A linear equation in two variables represent a straight line in *xy*
plane.

The father’s age is six
times his son’s age. Six years hence the age of father** **will be four times his
son’s age. Find the present ages (in years) of the son and father.

Let the present age of
father be* **x*** **years and the present age of son be

Given, *x* =
6*y
*… (1)

*x *+* *6* *=* *4(*y
*+* *6)
… (2)

Substituting (1) in (2), 6*y* + 6 = 4(*y*
+6)

6*y* + 6 = 4 *y*
+24 gives, *y*=9

Therefore, son’s age = 9
years and father’s age = 54 years.

Solve 2*x* − 3*y*
= 6 , *x* + *y* = 1

2*x* − 3*y* =
6 … (1)

*x *+* y *=* *1
… (2)

(1) × 1 gives, 2*x* - 3*y* = 6

(2) × 2 gives, 2*x* +2*y* = 2

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