**Linear equation in two variables**: It is the equation with two variables and with highest

power of variable or Degree .

**General form**:

$e.g.2x+3y=4,2a+3b=9$

** Solution of Linear equation in two variables**: The values of and which satisfy the given

equation is called solution of the equation in two variables.

e.g. 2x +3y+4=0 has solution x=1 and y =-2.

X= -5 , Y= 2 is also solution of the equation.

- The linear equation can have infinite solutions.
- The graph of linear equation is a straight line.
- The solution of a linear equation in variables is a point on the line graph.
- Any point on line is the solution of linear equation representing line.

**Pair of linear equations in 2 variables:**

It contains two linear equations in same two variables

$Ingeneralitiswrittenas{a}_{1}x+{b}_{1}y+{c}_{1}=0,{a}_{2}x+{b}_{2}y+{c}_{2}=0$

$with{a}_{1,}{b}_{1,}{a}_{2},{b}_{2}arerealand{{a}_{1}}^{2}+{{b}_{1}}^{2}\ne 0$

**Geometric meaning of Pair of linear equations in **** variables**:

$Givenpairofequations:{a}_{1}x+{b}_{1}y+{c}_{1}=0$

${a}_{2}x+{b}_{2}y+{c}_{2}=0$

**Case 1:**

When two lines intersect at a single point.

- There is unique solution.
- Equations are called consistent.

$\xb0\frac{{a}_{1}}{{a}_{2}}\ne \frac{{b}_{1}}{b{2}_{}}$

**Case 2:**

When two lines coincide with each other

- There are infinitely many solution
- Equations are called consistent

$\frac{{a}_{1}}{{a}_{2}}=\frac{{b}_{1}}{{b}_{2}}=\frac{{c}_{1}}{{c}_{2}}$

**Case 3:**

When two lines are parallel.

- No solution exist
- Equations are called inconsistent

$\frac{{a}_{1}}{{a}_{2}}=\frac{{b}_{1}}{{b}_{2}}\ne \frac{{c}_{1}}{{c}_{2}}$

**Algebraic method to solve ****pair of linear equations in **** variables**:

**Substitution method:**

- Find value of in terms of using any equation:

$e.g.y=\frac{{c}_{1}\u2013{a}_{1}x}{{b}_{1}}$

- Substitute this

${2}^{nd}$equation i.e. in

${a}_{2}x+{b}_{2}y+{c}_{2}=0$

- Find the value of x(or y)
- Put value x(or y) in eq in step 1.

**Elimination Method:**

- Make coefficients of x (or y) equal by multiplying with suitable non zero real number.
- Add or subtract both equations.
- Get the value of y( or x).
- Substitute value of y ( or x) in suitable equation and get value of other variable.

**Cross multiplication method:**

Arrange the coefficients as shown:

And we can write

$\frac{x}{{b}_{1}{c}_{2}\u2013{b}_{2}{c}_{1}}=\frac{y}{{c}_{1}a\u2013{c}_{2}{a}_{1}}=\frac{1}{{a}_{1}{b}_{2}\u2013{a}_{2}{b}_{1}}$

**Equations reducible to a pair linear equation in two variables:**

$Theequations\frac{{a}_{1}}{x}+\frac{{b}_{1}}{y}={c}_{1},\frac{{a}_{2}}{x}+\frac{{b}_{2}}{y}={c}_{2}$

$Substitute\frac{1}{x}=p,\frac{1}{y}=q$

$Andeq.sreduceto{a}_{1}p+{b}_{1}q+{c}_{1}=0\phantom{\rule{0ex}{0ex}}{a}_{2}p+{b}_{2}q+{c}_{2}=0$

This is pair of linear equations in variables and .

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