Question

If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of the equations is _______
(a) Inconsistent
(b) Having no solution
(c) Consistent
(d) None of the above

Answer 1

Hint:To solve this question, we will, first of all, define consistent and inconsistent solutions or the set of equations with examples. Then we will consider the given scenario and observer that for the given case which one or both or none of them is possible depending upon the definition of them.

Complete step by step answer:
Let us first of all define what consistent and inconsistent equations are.
Consistent Equation: A linear or non-linear system of equation is called consistent if there is at least one set of the values for the unknowns that satisfy each equation in the system that is when substituted each of the equations, they make each equation hold true as an identity.
Example: x + 2y = 14 and 2x + y = 6 is consistent as there is solution \[\left( x,y \right)=\left( \dfrac{-2}{3},\dfrac{22}{3} \right)\] satisfying both the equations.
Inconsistent equation: It is defined as one or more equations that are impossible to solve based on using one set of values for the variables.
Example: x + 2 = 4 and x + 2 = 6. If there is no one value of x satisfying solution x + 2 = 4 and x + 2 6, so there are inconsistent equations.
We will consider the question now. We are given that the linear intersect on a point. Let the lines be line 1 and line 2 means the solution is as depicted below.

If there is a common intersection there will be a value of the variables which satisfy both the equations of line 1 and of line 2. Therefore we will see that they are consistent as obvious by definition of consistent stated above. Hence, the pair of equations is consistent.
Hence, the option (c) is the right answer.

Note:
There is a chance to have more values of the set (x, y) satisfying both the equations. This is possible if we get any curve (not a straight line as two straight lines have one point of intersection).
Example: \[x\le 2,y\ge 0\] and \[x\le 5,y\ge 0\] have a lot more common point that only one. So, we can have more than one set of solutions.