Question

State whether each equation or function is linear. Write yes or no. If no, explain your reasoning.

$\frac{1}{x}+3y=-5$

Short answer

No, the equation $\frac{1}{x}+3y=-5$ is not a linear equation.

Step-by-step solution

– Definition of a linear equation and linear function.

In a linear equation the exponent of each variable is 1 and the variables may not be multiplied together. The graph of a linear equation is always a line.

In a linear function, each input value has a unique output value. In other words, for each point in domain, there is a unique value in the range of function. It can be written as $f\left(x\right)=mx+b$, where m and b are real numbers.

– Given information.

The given equation is:

$\frac{1}{x}+3y=-5$

Multiply both sides by x to rewrite the given equation.

localid="1647525013384" $\begin{array}{c}x(\frac{1}{x}+3y)=x(-5)\\ x\left(\frac{1}{x}\right)+x\left(3y\right)=-5x\\ 1+3xy=-5x\\ \end{array}$

– Check whether the equation is linear or not.

The equation $1+3xy=-5x$ is not a linear equation because the two variables are multiplied together.

The equations $\frac{1}{x}+3y=-5$ and $1+3xy=-5x$ are equivalent. So, the equation $\frac{1}{x}+3y=-5$ is not a linear equation.

Thus,the required answer is “No” because the equation $\frac{1}{x}+3y=-5$ is not a linear equation.