Question

Choose the correct option from the given options below by solving the following question:
The linear equation \[\left( {2x + 5y = 7} \right)\] has
A. A unique solution.
B. Two solutions.
C. Infinitely many solutions.
D. No solution.

Answer 1

Hint: In this question, we have a linear equation to find how many solutions it has? For that we have to identify the type of equation given and try substituting a few values to know the final answer.

Complete step-by-step solution:
Given equation,
\[2x + 5y = 7\]
The equation is a single linear equation which has two variables involved.
The two variables are correlated in the form of the given equation as a condition.
For every value of \[x\], we have a value of \[y\] which can satisfy the equation. Likewise, for every value of \[y\], we have a value of \[x\] which can satisfy the equation.
Let us take a few examples;
Going with the simple number \[1\], let us consider \[x = 1\].
Now, substituting \[x = 1\] in the equation provided, we can find the value of the other variable.
\[ \Rightarrow 2\left( 1 \right) + 5y = 7\]
Solving the linear equation above, we get;
\[ \Rightarrow 5y = 7 - 2\]
Subtracting the left-hand side, we get;
\[ \Rightarrow 5y = 5\]
Now dividing both the sides with the present constant to eliminate the same, we get;
\[\therefore y = 5\]
For a value of \[x\], we have a value of \[y\].
Similarly, let us consider a value for \[y\];
Taking \[y = 1\] and substituting it in the given equation, we get;
\[ \Rightarrow 2x + 5\left( 1 \right) = 7\]
Solving the linear equation above, we get;
\[ \Rightarrow 2x = 7 - 5\]
Subtracting the left-hand side, we get;
\[ \Rightarrow 2x = 2\]
Now dividing both the sides with the present constant to eliminate the same, we get;
\[\therefore x = 1\]
For a value of \[y\], we have a value of \[x\].
We have infinite numbers in the number systems. That implies for every infinite value of \[x\], we can have infinite values of \[y\].
Therefore, the equation has infinitely many solutions.

The correct option is C.

Note: This equation is a linear equation in two variables. A linear equation in two variables is represented in the form of \[ax + by + c = 0\] where, \[a,b\] and \[c\] are constants. Here, the degree of the equation is one which allows us to perform the normal algebraic operations without any complications of factorization and more.