Question

Which of the following equations is linear?
A. \[y = \dfrac{3}{x}\]
B. \[\sqrt x + y = 0\]
C. \[\dfrac{1}{2}x - \dfrac{5}{8}y = 11\]
D. \[y = {x^2} + 2x - 4\]

Answer 1

Hint: In the above given question, we are given four options which are different types of equations. We have to determine which one equation of the four given options is a linear equation. In order to approach the solution, first we have to check each equation if it satisfies the condition of being a linear equation or not.

Complete step by step answer:
Given that, four different equations of two variables \[x\] and \[y\]. We have to determine which one of them is a linear equation in two variables. First let us understand what makes a given equation to be called a linear equation.A linear equation can have any number of variables i.e. \[x\] , \[y\] and \[z\] . The only thing that makes them a linear equation is that each variable existing in the equation must have the highest power equal to unity i.e. \[1\]. In other words, the degree of a linear equation must always be \[1\]. A linear equation in two variables must be of the form \[ax + by + c = 0\] .

Now let us cross check each option whether they are linear equations or not.
A. \[y = \dfrac{3}{x}\]
We can also write is as
\[ \Leftarrow y - 3{x^{ - 1}} = 0\]
Hence it is not in the form of a linear equation.

B. \[\sqrt x + y = 0\]
This equation can be written as
\[ \Rightarrow {x^{\dfrac{1}{2}}} + y = 0\]
Hence it is not in the form of a linear equation.

C. \[\dfrac{1}{2}x - \dfrac{5}{8}y = 11\]
We can write it in the form of a linear equation as,
\[ \Rightarrow \dfrac{1}{2}x - \dfrac{5}{8}y - 11 = 0\]
Hence, it is a linear equation in two variables.

D. \[y = {x^2} + 2x - 4\]
This equation can be written as,
\[ \Rightarrow {x^2} + 2x - y - 4 = 0\]
Hence it is not in the form of a linear equation. Now, the only linear equation in the four given options is \[\dfrac{1}{2}x - \dfrac{5}{8}y = 11\].

Therefore, the correct option is C.

Note: Linear equations simply means that they are the equation of a straight line since the graphs of linear equations are linear i.e. a line. The standard equation of a straight line is given by \[y = mx + c\]. And we can write the linear equation \[ax + by + c = 0\] as,
\[ \Rightarrow by = - ax - c\]
\[ \Rightarrow y = - \dfrac{a}{b}x - \dfrac{c}{b}\]
\[ \Rightarrow y = mx + {c_0}\]
Which is the equation of a straight line where \[m = - \dfrac{a}{b}\] and \[{c_0} = - \dfrac{c}{b}\].