UpStudy (Fomerly CameraMath) - AI Homework Helper (2025)

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x=3y

Question

x=3y

Function

x=0

Evaluate

x=3y

\text{To find the }x\text{-intercept,set }y\text{=0}

x=3\times 0

Solution

x=0

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\text{Find the }x\text{-intercept/zero}

UpStudy (Fomerly CameraMath) - AI Homework Helper (4)

Find the y-intercept

UpStudy (Fomerly CameraMath) - AI Homework Helper (5)

Find the slope

UpStudy (Fomerly CameraMath) - AI Homework Helper (6)

Solve the equation

y=\frac{x}{3}

Evaluate

x=3y

Swap the sides of the equation

3y=x

\text{Multiply both sides of the equation by }\frac{1}{3}

3y\times \frac{1}{3}=x\times \frac{1}{3}

Calculate the product

3y\times \frac{1}{3}=\frac{x}{3}

Solution

y=\frac{x}{3}

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Testing for symmetry

\textrm{Symmetry with respect to the origin}

Evaluate

x=3y

\text{To test if the graph of }x=3y\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}

-x=3\left(-y\right)

Evaluate

-x=-3y

Solution

\textrm{Symmetry with respect to the origin}

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Testing for symmetry about the origin

UpStudy (Fomerly CameraMath) - AI Homework Helper (7)

Testing for symmetry about the x-axis

UpStudy (Fomerly CameraMath) - AI Homework Helper (8)

Testing for symmetry about the y-axis

UpStudy (Fomerly CameraMath) - AI Homework Helper (9)

Rewrite the equation

\begin{align}&r=0\\&\theta =\arctan\left(\frac{1}{3}\right)+k\pi ,k \in \mathbb{Z}\end{align}

Evaluate

x=3y

Move the expression to the left side

x-3y=0

\text{To convert the equation to polar coordinates,substitute }x\text{ for }r\cos\left(\theta \right)\text{ and }y\text{ for }r\sin\left(\theta \right)

\cos\left(\theta \right)\times r-3\sin\left(\theta \right)\times r=0

Factor the expression

\left(\cos\left(\theta \right)-3\sin\left(\theta \right)\right)r=0

Separate into possible cases

\begin{align}&r=0\\&\cos\left(\theta \right)-3\sin\left(\theta \right)=0\end{align}

Solution

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Evaluate

\cos\left(\theta \right)-3\sin\left(\theta \right)=0

Move the expression to the right side

-3\sin\left(\theta \right)=0-\cos\left(\theta \right)

Subtract the terms

-3\sin\left(\theta \right)=-\cos\left(\theta \right)

Divide both sides

\frac{-3\sin\left(\theta \right)}{\cos\left(\theta \right)}=-1

Divide the terms

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Evaluate

\frac{-3\sin\left(\theta \right)}{\cos\left(\theta \right)}

Rewrite the expression

-\frac{3\sin\left(\theta \right)}{\cos\left(\theta \right)}

Rewrite the expression

-3\cos^{-1}\left(\theta \right)\sin\left(\theta \right)

Rewrite the expression

-3\tan\left(\theta \right)

-3\tan\left(\theta \right)=-1

\text{Multiply both sides of the equation by }-\frac{1}{3}

-3\tan\left(\theta \right)\left(-\frac{1}{3}\right)=-\left(-\frac{1}{3}\right)

Calculate

\tan\left(\theta \right)=-\left(-\frac{1}{3}\right)

Multiplying or dividing an even number of negative terms equals a positive

\tan\left(\theta \right)=\frac{1}{3}

Use the inverse trigonometric function

\theta =\arctan\left(\frac{1}{3}\right)

\text{Add the period of }k\pi ,k \in \mathbb{Z}\text{ to find all solutions}

\theta =\arctan\left(\frac{1}{3}\right)+k\pi ,k \in \mathbb{Z}

\begin{align}&r=0\\&\theta =\arctan\left(\frac{1}{3}\right)+k\pi ,k \in \mathbb{Z}\end{align}

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Rewrite in polar form

UpStudy (Fomerly CameraMath) - AI Homework Helper (10)

Rewrite in standard form

UpStudy (Fomerly CameraMath) - AI Homework Helper (11)

Rewrite in slope-intercept form

UpStudy (Fomerly CameraMath) - AI Homework Helper (12)

Find the first derivative

\frac{dy}{dx}=\frac{1}{3}

Calculate

x=3y

Take the derivative of both sides

\frac{d}{dx}\left(x\right)=\frac{d}{dx}\left(3y\right)

\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}

1=\frac{d}{dx}\left(3y\right)

Calculate the derivative

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Evaluate

\frac{d}{dx}\left(3y\right)

Use differentiation rules

\frac{d}{dy}\left(3y\right)\times \frac{dy}{dx}

Evaluate the derivative

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Evaluate

\frac{d}{dy}\left(3y\right)

\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))

3\times \frac{d}{dy}\left(y\right)

\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}

3\times 1

Any expression multiplied by 1 remains the same

3

3\frac{dy}{dx}

1=3\frac{dy}{dx}

Swap the sides of the equation

3\frac{dy}{dx}=1

\text{Multiply both sides of the equation by }\frac{1}{3}

3\frac{dy}{dx}\times \frac{1}{3}=1\times \frac{1}{3}

Any expression multiplied by 1 remains the same

3\frac{dy}{dx}\times \frac{1}{3}=\frac{1}{3}

Solution

\frac{dy}{dx}=\frac{1}{3}

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\text{Find the derivative with respect to }x

UpStudy (Fomerly CameraMath) - AI Homework Helper (13)

\text{Find the derivative with respect to }y

UpStudy (Fomerly CameraMath) - AI Homework Helper (14)

Find the second derivative

\frac{d^2y}{dx^2}=0

Calculate

x=3y

Take the derivative of both sides

\frac{d}{dx}\left(x\right)=\frac{d}{dx}\left(3y\right)

\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}

1=\frac{d}{dx}\left(3y\right)

Calculate the derivative

More Steps Hide Steps

Evaluate

\frac{d}{dx}\left(3y\right)

Use differentiation rules

\frac{d}{dy}\left(3y\right)\times \frac{dy}{dx}

Evaluate the derivative

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Evaluate

\frac{d}{dy}\left(3y\right)

\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))

3\times \frac{d}{dy}\left(y\right)

\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}

3\times 1

Any expression multiplied by 1 remains the same

3

3\frac{dy}{dx}

1=3\frac{dy}{dx}

Swap the sides of the equation

3\frac{dy}{dx}=1

\text{Multiply both sides of the equation by }\frac{1}{3}

3\frac{dy}{dx}\times \frac{1}{3}=1\times \frac{1}{3}

Any expression multiplied by 1 remains the same

3\frac{dy}{dx}\times \frac{1}{3}=\frac{1}{3}

\text{Cancel out the greatest common factor }3

\frac{dy}{dx}=\frac{1}{3}

Take the derivative of both sides

\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{1}{3}\right)

Calculate the derivative

\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{1}{3}\right)

Solution

\frac{d^2y}{dx^2}=0

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\text{Find the derivative with respect to }x

UpStudy (Fomerly CameraMath) - AI Homework Helper (15)

\text{Find the derivative with respect to }y

UpStudy (Fomerly CameraMath) - AI Homework Helper (16)

Graph

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