We are offering TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables to learn maths more effectively.

## TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables

→ An Equation of the form ax + by + c = 0, where a, b, c e R and a and b are not both zero, is called a linear equation in two variables x and y.

→ A pair of linear equations in two variables x and y can be represented as follows :

a_{1}x + b_{1}y + C_{1} = 0

a_{2}x + b_{2}y + c_{2} = 0

Where a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2} are real numbers such that a_{1}^{2} + b_{1}^{2} ≠ 0; a_{2}^{2} + b_{2}^{2} ≠ 0.

→ A pair of linear Equations in two variables forms a system of simultaneous linear equations.

Example : 3x – 4y = 2

2x + 5y = 9

→ A pair of values of the variables x and y satisfying each one of the equations that are given is called a solution of the system.

x = 2, y = 1 is a solution of the system of simultaneous linear equations.

3x-4y = 2 …………… (1)

2x + 5y = 9 …………… (2)

Putting x = 2 and y = 1 in equation (1), we get

L.H.S. = 3 × 2 – 4 × 1 = 6 – 4 = 2

R.H.S = 2

L.H.S = R.H.S

Similarly, put x = 2 and y = 1 in equation (2), we get

L.H.S = 2 × 2 + 5 × 1 = 4 + 5 = 9

R.H.S = 9

∴ L.H.S = R.H.S

→ A pair of linear equations in two variables can be solved using

- Graphical method
- Model method
- Algebraic method :

(a) Substitution method

(b) Elimination method

(c) Cross-Multiplication method

→ Graphical method: The graph of a pair of linear equations in two variables is represented by two straight lines.

- If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.
- If the lines coincide, then there are infinitely many solutions – each point on the line being a solution. In this case, the pair of equations is dependent (consistent).
- If the lines are parallel, then the pair of equations has no solution.

In this case, the pair of equations is inconsistent.

→ The relation that exists between the coefficients and nature of system of equations.

→ If a pair of linear equations is given by a_{1}x + b_{2}y + c_{2} = 0 and a_{2}x + b_{2}y + c_{2} = 0, then

- The pair of linear equations is consistent if \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
- The pair of linear equation is inconsistent if \(\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
- The pair of linear equation is dependent and consistent if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)

Important Formulas:

- The pair of linear equations is consistent if \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
- The pair of linear equation is inconsistent if \(\frac{a_1}{a_2}=\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
- The pair of linear equation is dependent and consistent if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)

Flow Chat:

William George Horner(1786 – 1832):

- A William George Horner was a British mathematician; he was a schoolmaster; headmaster and schoolkeeper, proficient in classics as well as mathematics.
- A He wrote extensively on functional equations, number theory and approximation theory.
- A His contribution to approximation theory is honoured in the designation Horner’s method. The modern invention of the zoetrope, under the name Daedaleum in 1834 has been attributed to him.