Question

The linear equation \[2x + 5y = 7\] has
A) has an unique solution.
B) two solutions.
C) infinitely many solutions.
D) no solution.

Answer 1

Hint:
This question is based on the concept of linear equations. First separate either of the variables \[x\] or \[y\] to one side of the equation and all the other terms to the other side. Then try to find the number of values of \[x\] and \[y\] satisfying the equation.

Complete step by step solution:
Linear equations are equations of the first order that are defined for lines in the coordinate system. We have to find the number of possible solutions of the given linear equation i.e. \[2x + 5y = 7\].
First we separate \[x\] to one side of the equation:
\[2x + 5y = 7\]
\[ \Rightarrow 2x = 7 - 5y\]
Dividing both sides by \[2\], we get
\[ \Rightarrow x = \dfrac{{7 - 5y}}{2}\]
Now we can substitute \[y\] with any number to get the corresponding value of \[x\]. For example substituting \[y = 1\], \[x = 1\] similarly substituting \[y = 2\], \[x = \dfrac{{ - 3}}{2}\].
Similarly there will be infinite values of \[x\] for infinite values of \[y\]. Thus there will be infinite solutions of the given equations.

So the correct option is (c).

Note:
This problem can also be solved geometrically also. Any linear equation represents a line. We know there are infinite points on a line. Each of these points will have a specific coordinate. Since the entire line is represented by the equation, therefore all the points on the line will satisfy it. Thus there will be infinite solutions.