Solve the following pair of linear equations by graphical method and find the nature of roots.

x + y = 5; 2x + 2y = 10

For a given pair of linear equations in two variables, the graph is represented by two lines.

● If the lines intersect at a point, that point gives the unique solution for the two equations. If there is a unique solution of the given pair of equations, the equations are called consistent.

● If the lines coincide, there are indefinitely many solutions for the pair of linear equations. In this case, each point on the line is a solution. If there are infinitely many solutions of the given pair of linear equations, the equations are called dependent (consistent).

● If the lines are parallel, there is no solution for the pair of linear equations. If there is no solution of the given pair of linear equations, the equations are called inconsistent.

To plot the given pair of linear equations, we substitute x = 0 or y = 0

in the given linear equations to get x and y. To find more points on the

lines, take different values of x related to it and we get different values

for y from the equation.

We get the following tables for the given linear equations.

For x + y = 5

y = 5 – x

For 2x + 2y = 10

2y = 10 – 2x

As seen from the graph and the table, the two lines are parallel and

overlap each other. Hence, roots of the equations are consistent or

dependent with infinitely many solutions.