Pair of Linear Equations in Two Variables

An equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables x and y. The graph of a linear equation in two variables is always a straight line.

Two linear equations in the same two variables are called a pair of linear equations in two variables. The general form of a pair of linear equations in two variables x and y is:\na₁x + b₁y + c₁ = 0\na₂x + b₂y + c₂ = 0\nwhere a₁, b₁, c₁, a₂, b₂, c₂ are real numbers such that a₁² + b₁² ≠ 0 and a₂² + b₂² ≠ 0.

To solve a pair of linear equations graphically, we plot the graph of each equation. Each equation represents a straight line. The solution(s) of the pair of linear equations is given by the point(s) of intersection of these lines.\n\nThere are three possibilities:\n1. **Intersecting Lines**: The two lines intersect at a single point. In this case, the pair of equations has a unique solution. Such a pair of equations is called a consistent pair.\n Condition: a₁/a₂ ≠ b₁/b₂\n2. **Coincident Lines**: The two lines are identical (one lies exactly over the other). In this case, there are infinitely many solutions, as every point on the line is a common solution. Such a pair of equations is called a consistent and dependent pair.\n Condition: a₁/a₂ = b₁/b₂ = c₁/c₂\n3. **Parallel Lines**: The two lines are parallel and do not intersect at any point. In this case, there is no solution. Such a pair of equations is called an inconsistent pair.\n Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

This method involves expressing one variable in terms of the other from one equation and substituting this expression into the second equation. This reduces the pair of linear equations to a single linear equation in one variable, which can then be solved.\n\nSteps:\n1. Choose either of the two equations and express one variable (say, y) in terms of the other variable (say, x).\n2. Substitute this expression for y in the other equation. This will give a linear equation in a single variable x.\n3. Solve the equation for x.\n4. Substitute the value of x obtained in Step 3 into the expression for y (from Step 1) to find the value of y.

In this method, we eliminate one of the variables by making its coefficients numerically equal in both equations and then adding or subtracting the equations.\n\nSteps:\n1. Multiply both equations by some suitable non-zero constants (if necessary) to make the coefficients of one variable (either x or y) numerically equal.\n2. Add or subtract one equation from the other so that one variable gets eliminated.\n3. Solve the resulting equation in one variable.\n4. Substitute the value of this variable in either of the original equations to find the value of the other variable.

This method provides a direct formula for finding the solution of a pair of linear equations. It is particularly useful for finding a unique solution.\n\nConsider the pair of linear equations:\na₁x + b₁y + c₁ = 0\na₂x + b₂y + c₂ = 0\n\nThe solution (x, y) can be found using the following formula, provided a₁b₂ - a₂b₁ ≠ 0 (i.e., unique solution case):