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Introduction

Pair of Linear Equations in two Variables

A variable is a number which is use in alphanumeric character. e.g., x, y, a, b etc.

A non-variable or constant is a number which is use in numeric values. e.g., 2, 3, 10 etc.

Each solution (x, y) of a linear equation in two variables, ax + by + c = 0. Equations like this are called a pair of linear equations in two variables. e.g., 2x + 3y – 7 = 0 and 9x – 2y + 8 = 0.

(i) The two lines will intersect at one point. e.g., 2x + 3y – 7 = 0 and 9x – 2y + 8 = 0.

(ii) The two lines will not intersect, i.e., they are parallel. e.g., 2x + 3y – 7 = 0 and 4x + 6y - 14 = 0.

(iii) The two lines will be coincident. e.g., 2x + 3y – 7 = 0 and 2x + 3y - 7 = 0.

A pair of linear equations in two variables, which has a solution, is called a consistent pair of linear equations.

A pair of linear equations which has no solution, is called an inconsistent pair of linear equations.

A pair of linear equations which are equivalent has infinitely many distinct common solutions. Such a pair is called a Dependent pair of linear equations in two variables.

(i) Substitution Method

(ii) Elimination Method

(iii) Cross-Multiplication Method

**Step 1.** Find the value of one variable, say y in terms of the other variable, i.e., x from either equation, whichever is convenient.

**Step 2.** Substitute this value of y in the other equation, and reduce it to an equation in one variable, i.e., in terms of x, which can be solved.
Sometimes, as in Examples 9 and 10 below, you can get statements with no variable.
If this statement is true, you can conclude that the pair of linear equations has infinitely many solutions.
If the statement is false, then the pair of linear equations is inconsistent.

**Step 3.** Substitute the value of x (or y) obtained in Step 2 in the equation used in Step 1 to obtain the value of the other variable.

**Step 1.** Firstly multiply both the equations by some suitable non-zero constants to make the coefficients
of one variable (either x or y) numerically equal.

**Step 2.** Then add or subtract one equation from the other so that one variable gets eliminated.
If you get an equation in one variable, go to Step 3.
If in Step 2, we obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions.
If in Step 2, we obtain a false statement involving no variable, then the original pair of equations has no solution, i.e., it is inconsistent.

**Step 3.** Solve the equation in one variable (x or y) so obtained to get its value.

**Step 4.** Substitute this value of x (or y) in either of the original equations to get the value of the other variable.

a₁x + b₁y + c₁ = 0 ....... (i) and
a₂x + b₂y + c₂ = 0 ........(ii)
**Step 1.** Write the given equations in the form (i) and (ii).

**Step 2.** Taking the help of the diagram above, write Equations as given in (8).

**Step 3.** Find x and y, provided a₁b₂ – a₂b₁ ≠ 0

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