Question

Which equation is non-linear?

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

Answer 1

Answer:

$\huge\boxed{\text{A is not linear.}}$

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 $y=mx+b$ when put in that order.

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

• Linear = has a constant rate of change
• Exponential = increases for x as the exponent
• Quadratic = is in a binomial, trinomial, or polynomial -  it has a vertex

Linear functions are usually in the form $y=mx+b$

Exponential functions are usually in the form $y=b^x$

Quadratic functions are usually in the form $ax^2 + bx + c$

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 $y=mx+b$ ($-3x--7y=-21$ can be simplified to $y= \frac{3}{7}x-3$)

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 2

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:

• Linear equations cannot have an exponent.
• Linear equations have a constant slope and are straight lines.
• Linear equations can be negative or positive.
• Linear equations have a domain of all real numbers. They also have a range of all real numbers.

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

Equation A

$\displaystyle y = 3x^3 - 10$

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

$\displaystyle y = \frac{1}{2}x + 4$

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

$\displaystyle y = 4x -\frac{5}{3}$

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

$-3x--7y = -21$

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

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

Our final equation is $\displaystyle y = \frac{3}{7}x-3$.

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:

• Equation A: $\displaystyle y = 3x^3 - 10 \ \text{X}$
• Equation B: $\displaystyle y = \frac{1}{2}x + 4 \ \checkmark$
• Equation C: $\displaystyle y = 4x -\frac{5}{3} \ \checkmark$
• Equation D: $\displaystyle y = \frac{3}{7}x-3 \ \checkmark$

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