Solve 2y-3x=0 - ScanMath

Question

Function

Find the x -intercept/zero

Find the y-intercept

Find the slope

x = 0

Evaluate

2 y - 3 x = 0

To find the x -intercept,set y =0

2 \times 0 - 3 x = 0

Any expression multiplied by 0 equals 0

0 - 3 x = 0

Removing 0 doesn't change the value,so remove it from the expression

- 3 x = 0

Change the signs on both sides of the equation

3 x = 0

Solution

x = 0

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Solve the equation

Solve for x

Solve for y

x = \frac{2 y}{3}

Evaluate

2 y - 3 x = 0

Move the expression to the right-hand side and change its sign

- 3 x = 0 - 2 y

Removing 0 doesn't change the value,so remove it from the expression

- 3 x = - 2 y

Change the signs on both sides of the equation

3 x = 2 y

Divide both sides

\frac{3 x}{3} = \frac{2 y}{3}

Solution

x = \frac{2 y}{3}

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Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Symmetry with respect to the origin

Evaluate

2 y - 3 x = 0

To test if the graph of 2 y - 3 x = 0 is symmetry with respect to the origin,substitute -x for x and -y for y

2 (- y) - 3 (- x) = 0

Evaluate

More Steps

Evaluate

2 (- y) - 3 (- x)

Multiply the numbers

- 2 y - 3 (- x)

Multiply the numbers

- 2 y - (- 3 x)

Rewrite the expression

- 2 y + 3 x

- 2 y + 3 x = 0

Solution

Symmetry with respect to the origin

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Rewrite the equation

Rewrite in polar form

Rewrite in standard form

Rewrite in slope-intercept form

r = 0 θ = arccot (\frac{2}{3}) + kπ, k \in Z

Evaluate

2 y - 3 x = 0

To convert the equation to polar coordinates,substitute x for r cos (θ) and y for r sin (θ)

2 sin (θ) \times r - 3 cos (θ) \times r = 0

Factor the expression

(2 sin (θ) - 3 cos (θ)) r = 0

Separate into possible cases

r = 0 2 sin (θ) - 3 cos (θ) = 0

Solution

More Steps

Evaluate

2 sin (θ) - 3 cos (θ) = 0

Move the expression to the right side

- 3 cos (θ) = 0 - 2 sin (θ)

Subtract the terms

- 3 cos (θ) = - 2 sin (θ)

Divide both sides

\frac{- 3 cos ( θ )}{sin ( θ )} = - 2

Divide the terms

More Steps

Evaluate

\frac{- 3 cos ( θ )}{sin ( θ )}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{3 cos ( θ )}{sin ( θ )}

Rewrite the expression

- 3 sin^{- 1} (θ) cos (θ)

Rewrite the expression

- 3 cot (θ)

- 3 cot (θ) = - 2

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

- 3 cot (θ) (- \frac{1}{3}) = - 2 (- \frac{1}{3})

Calculate

cot (θ) = - 2 (- \frac{1}{3})

Calculate

More Steps

Evaluate

- 2 (- \frac{1}{3})

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

2 \times \frac{1}{3}

Multiply the numbers

\frac{2}{3}

cot (θ) = \frac{2}{3}

Use the inverse trigonometric function

θ = arccot (\frac{2}{3})

Add the period of kπ, k \in Z to find all solutions

θ = arccot (\frac{2}{3}) + kπ, k \in Z

r = 0 θ = arccot (\frac{2}{3}) + kπ, k \in Z

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = \frac{3}{2}

Calculate

2 y - 3 x = 0

Take the derivative of both sides

\frac{d}{d x} (2 y - 3 x) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 y - 3 x)

Use differentiation rules

\frac{d}{d x} (2 y) + \frac{d}{d x} (- 3 x)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 y)

Use differentiation rules

\frac{d}{d y} (2 y) \times \frac{d y}{d x}

Evaluate the derivative

2 \frac{d y}{d x}

2 \frac{d y}{d x} + \frac{d}{d x} (- 3 x)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 3 x)

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

- 3 \times \frac{d}{d x} (x)

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

- 3 \times 1

Any expression multiplied by 1 remains the same

- 3

2 \frac{d y}{d x} - 3

2 \frac{d y}{d x} - 3 = \frac{d}{d x} (0)

Calculate the derivative

2 \frac{d y}{d x} - 3 = 0

Move the constant to the right-hand side and change its sign

2 \frac{d y}{d x} = 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

2 \frac{d y}{d x} = 3

Divide both sides

\frac{2 \frac{d y}{d x}}{2} = \frac{3}{2}

Solution

\frac{d y}{d x} = \frac{3}{2}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

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

Calculate

2 y - 3 x = 0

Take the derivative of both sides

\frac{d}{d x} (2 y - 3 x) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 y - 3 x)

Use differentiation rules

\frac{d}{d x} (2 y) + \frac{d}{d x} (- 3 x)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (2 y)

Use differentiation rules

\frac{d}{d y} (2 y) \times \frac{d y}{d x}

Evaluate the derivative

2 \frac{d y}{d x}

2 \frac{d y}{d x} + \frac{d}{d x} (- 3 x)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 3 x)

Use differentiation rule \frac{d}{d x} (c f (x)) = c \times \frac{d}{d x} (f (x))

- 3 \times \frac{d}{d x} (x)

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

- 3 \times 1

Any expression multiplied by 1 remains the same

- 3

2 \frac{d y}{d x} - 3

2 \frac{d y}{d x} - 3 = \frac{d}{d x} (0)

Calculate the derivative

2 \frac{d y}{d x} - 3 = 0

Move the constant to the right-hand side and change its sign

2 \frac{d y}{d x} = 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

2 \frac{d y}{d x} = 3

Divide both sides

\frac{2 \frac{d y}{d x}}{2} = \frac{3}{2}

Divide the numbers

\frac{d y}{d x} = \frac{3}{2}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{3}{2})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{3}{2})

Solution

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

Show Solution

Function

Solve the equation

Testing for symmetry

Rewrite the equation

Find the first derivative

Find the second derivative

Graph