Solve 3x-5y=0 - ScanMath

Question

Function

Find the x -intercept/zero

Find the y-intercept

Find the slope

x = 0

Evaluate

3 x - 5 y = 0

To find the x -intercept,set y =0

3 x - 5 \times 0 = 0

Any expression multiplied by 0 equals 0

3 x - 0 = 0

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

3 x = 0

Solution

x = 0

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Solve the equation

Solve for x

Solve for y

x = \frac{5 y}{3}

Evaluate

3 x - 5 y = 0

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

3 x = 0 + 5 y

Add the terms

3 x = 5 y

Divide both sides

\frac{3 x}{3} = \frac{5 y}{3}

Solution

x = \frac{5 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

3 x - 5 y = 0

To test if the graph of 3 x - 5 y = 0 is symmetry with respect to the origin,substitute -x for x and -y for y

3 (- x) - 5 (- y) = 0

Evaluate

More Steps

Evaluate

3 (- x) - 5 (- y)

Multiply the numbers

- 3 x - 5 (- y)

Multiply the numbers

- 3 x - (- 5 y)

Rewrite the expression

- 3 x + 5 y

- 3 x + 5 y = 0

Solution

Symmetry with respect to the origin

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Rewrite the equation

Rewrite in polar form

Rewrite in slope-intercept form

r = 0 θ = arctan (\frac{3}{5}) + kπ, k \in Z

Evaluate

3 x - 5 y = 0

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

3 cos (θ) \times r - 5 sin (θ) \times r = 0

Factor the expression

(3 cos (θ) - 5 sin (θ)) r = 0

Separate into possible cases

r = 0 3 cos (θ) - 5 sin (θ) = 0

Solution

More Steps

Evaluate

3 cos (θ) - 5 sin (θ) = 0

Move the expression to the right side

- 5 sin (θ) = 0 - 3 cos (θ)

Subtract the terms

- 5 sin (θ) = - 3 cos (θ)

Divide both sides

\frac{- 5 sin ( θ )}{cos ( θ )} = - 3

Divide the terms

More Steps

Evaluate

\frac{- 5 sin ( θ )}{cos ( θ )}

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

- \frac{5 sin ( θ )}{cos ( θ )}

Rewrite the expression

- 5 cos^{- 1} (θ) sin (θ)

Rewrite the expression

- 5 tan (θ)

- 5 tan (θ) = - 3

Multiply both sides of the equation by - \frac{1}{5}

- 5 tan (θ) (- \frac{1}{5}) = - 3 (- \frac{1}{5})

Calculate

tan (θ) = - 3 (- \frac{1}{5})

Calculate

More Steps

Evaluate

- 3 (- \frac{1}{5})

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

3 \times \frac{1}{5}

Multiply the numbers

\frac{3}{5}

tan (θ) = \frac{3}{5}

Use the inverse trigonometric function

θ = arctan (\frac{3}{5})

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

θ = arctan (\frac{3}{5}) + kπ, k \in Z

r = 0 θ = arctan (\frac{3}{5}) + 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}{5}

Calculate

3 x - 5 y = 0

Take the derivative of both sides

\frac{d}{d x} (3 x - 5 y) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 x - 5 y)

Use differentiation rules

\frac{d}{d x} (3 x) + \frac{d}{d x} (- 5 y)

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

3 + \frac{d}{d x} (- 5 y)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 5 y)

Use differentiation rules

\frac{d}{d y} (- 5 y) \times \frac{d y}{d x}

Evaluate the derivative

- 5 \frac{d y}{d x}

3 - 5 \frac{d y}{d x}

3 - 5 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

3 - 5 \frac{d y}{d x} = 0

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

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

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

- 5 \frac{d y}{d x} = - 3

Change the signs on both sides of the equation

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

Divide both sides

\frac{5 \frac{d y}{d x}}{5} = \frac{3}{5}

Solution

\frac{d y}{d x} = \frac{3}{5}

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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

3 x - 5 y = 0

Take the derivative of both sides

\frac{d}{d x} (3 x - 5 y) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 x - 5 y)

Use differentiation rules

\frac{d}{d x} (3 x) + \frac{d}{d x} (- 5 y)

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

3 + \frac{d}{d x} (- 5 y)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 5 y)

Use differentiation rules

\frac{d}{d y} (- 5 y) \times \frac{d y}{d x}

Evaluate the derivative

- 5 \frac{d y}{d x}

3 - 5 \frac{d y}{d x}

3 - 5 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

3 - 5 \frac{d y}{d x} = 0

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

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

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

- 5 \frac{d y}{d x} = - 3

Change the signs on both sides of the equation

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

Divide both sides

\frac{5 \frac{d y}{d x}}{5} = \frac{3}{5}

Divide the numbers

\frac{d y}{d x} = \frac{3}{5}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{3}{5})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{3}{5})

Solution

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

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Function

Solve the equation

Testing for symmetry

Rewrite the equation

Find the first derivative

Find the second derivative

Graph