Which equation is non-linear?

A) [tex]y=3x^3-10[/tex]
B) [tex]y=1/2x+4[/tex]
C) [tex]y=4x-5/3[/tex]
D) [tex]-3x--7y=-21[/tex]

Answer:

[tex]\huge\boxed{\text{A is not linear.}}[/tex]

Step-by-step explanation:

Linear equations are equations that, when graphed, form a "line" and have a constant rate of change.

Usually, these are in the form [tex]y=mx+b[/tex] when put in that order.

There are a few types of equations, I'll list some of the major ones

Linear functions are usually in the form [tex]y=mx+b[/tex]

Exponential functions are usually in the form [tex]y=b^x[/tex]

Quadratic functions are usually in the form [tex]ax^2 + bx + c[/tex]

We must note that linear functions can not

A) Have an exponent on any x term

B) Have x and y multiplied by each other

We can see that every equation, except A

A) Follows the form [tex]y=mx+b[/tex] ([tex]-3x--7y=-21[/tex] can be simplified to [tex]y= \frac{3}{7}x-3[/tex])

B) Does not have any exponents on their x terms

That leaves A) to be the equation that isn't linear.

Hope this helped!

Answer:

Equation A is non-linear.

Step-by-step explanation:

We are given four equations in which we need to determine if they are linear or non-linear relationships.

Firstly, we need to know some information about linear equations:

Now, in order to check this easily, we need to place these equations into slope-intercept form.

Equation A

[tex]\displaystyle y = 3x^3 - 10[/tex]

We see that this has an exponent, so this cannot be a linear equation.

In fact, because it is cubed, this is called a cubic function. Therefore, equation A is not a linear equation.

Equation B

[tex]\displaystyle y = \frac{1}{2}x + 4[/tex]

This equation does not have a exponent. It also has a slope and a y-intercept. Therefore, equation B is a linear equation.

Equation C

[tex]\displaystyle y = 4x -\frac{5}{3}[/tex]

The equation does not have a exponent. It also has a slope and a y-intercept. Therefore, equation C is a linear equation.

Equation D

[tex]-3x--7y = -21[/tex]

First off, we need to get this equation into slope-intercept form, if possible.

[tex]\displaystyle -3x -- 7y = -21\\\\-3x + 7y = -21 \ \ \ \text{Change the sign on 7y.}\\\\7y = 3x - 21 \ \ \ \text{Move 3x to the other side by adding.}\\\\\frac{7y}{7}=\frac{3x-21}{7} \ \ \ \text{Divide both sides by 7 to isolate y.}\\\\\bold{y = \frac{3}{7}x-3}[/tex]

Our final equation is [tex]\displaystyle y = \frac{3}{7}x-3[/tex].

The equation does not have an exponent. It has a constant slope and a y-intercept. Therefore, equation D is a linear equation.

Let's check our equations:

Therefore, we have determined that Equation A is not a linear equation, making it non-linear.

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