Question

A linear equation in two variable has maximum
A. Only one solution
B. Two solutions
C. Infinite solutions
D. None of these

Answer 1

Hint: A linear equation of two variables represents a straight line so the number of points containing on it represent the number of solutions.

Complete step-by-step answer:
In the question, we are asked how many maximum solutions a linear equation in two variables will have. So, before answering this, let us find out what a linear equation is. In mathematics, a linear equation is an equation that can be put in the form ${{a}_{1}}{{x}_{1}}+{{a}_{2}}{{x}_{2}}+\ldots +{{a}_{n}}{{x}_{n}}+b=0$, where ${{x}_{1,}}\ldots ,{{x}_{n}}$ are the variables and $b,{{a}_{1}},{{a}_{2}},\ldots ,{{a}_{n}}$ are the coefficients which are often real numbers. The coefficients are considered as parameters of the equation and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaning to the equation, the coefficients required must be 0.
Alternatively, a linear equation can be obtained by equating to a 0 and a linear polynomial over some field from which the coefficients are taken. Linear equations occur frequently in all fields of mathematics and their application occurs in physics and engineering partly because nonlinear systems are well approximated by linear equations. More generally, the solutions of a linear equation in n variables form a hyperbola (a subspace of dimension n - 1) in the Euclidean space of dimensions. The solutions of such an equation are the values that when substituted for the unknowns, make the equality true. In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane.
A linear equation in two variables has a maximum of infinite solutions.
Therefore, the correct answer is option C.

Note: A linear equation of two variables represents a straight line containing infinite number of points, each representing a solution, thus confirming the fact that a linear equation in two variables has maximum infinite solutions. Most often the students choose the answer as two. It is because they think that there will be two solutions for two variables, but it is not valid here. They might be confused with the case of quadratic equations, where we will obtain 2 solutions for a single variable. So, this must be kept in mind.