Chapter 1.CR, Problem 1CR

Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables $x$ and $y.$

$2x-y^2=4$

To determine

Whether the equation $2x-y^2=4$ is linear in the variables x and y or not.

Answer to Problem 1CR

Solution:

The equation $2x-y^2=4$ is not linear in y.

Explanation of Solution

Given:

The provided equation is $2x-y^2=4$.

Definition used:

Linear Equation in n variables:

A linear equation in n variables $x_1,x_2,x_3,\mathrm{...},x_n$ has the form $a_1{x}_1+a_2{x}_2+a_3{x}_3+\mathrm{...}+a_n{x}_n=b$. The coefficients $a_1,a_2,a_3,\mathrm{...},a_n$ are real numbers, and the constant term b is a real number. The number $a_1$ is the leading coefficient, and $x_1$ is the leading variable.

Calculation:

Consider $2x-y^2=4$.

We know that the linear equation is of the form $a_1{x}_1+a_2{x}_2+a_3{x}_3+\mathrm{...}+a_n{x}_n=b$ where $a_1,a_2,a_3,\mathrm{...},a_n$ are coefficients and b is the constant.

The given equation is $2x-y^2=4$.

Here we see that the power of y is 2.

Thus the given equation is not linear in y.

Final statement:

Therefore the equation $2x-y^2=4$ is not linear in y.