Solve 3x - 5y 12 = 0

Question

Function

Find the x -intercept/zero

Find the y-intercept

Find the slope

x = 0

Evaluate

3 x - 5 y \times 12 = 0

To find the x -intercept,set y =0

3 x - 5 \times 0 \times 12 = 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 = 20 y

Evaluate

3 x - 5 y \times 12 = 0

Multiply the terms

3 x - 60 y = 0

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

3 x = 0 + 60 y

Add the terms

3 x = 60 y

Divide both sides

\frac{3 x}{3} = \frac{60 y}{3}

Divide the numbers

x = \frac{60 y}{3}

Solution

More Steps

Evaluate

\frac{60 y}{3}

Reduce the numbers

\frac{20 y}{1}

Calculate

20 y

x = 20 y

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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 12 = 0

Simplify the expression

3 x - 60 y = 0

To test if the graph of 3 x - 60 y = 0 is symmetry with respect to the origin,substitute -x for x and -y for y

3 (- x) - 60 (- y) = 0

Evaluate

More Steps

Evaluate

3 (- x) - 60 (- y)

Multiply the numbers

- 3 x - 60 (- y)

Multiply the numbers

- 3 x - (- 60 y)

Rewrite the expression

- 3 x + 60 y

- 3 x + 60 y = 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 = arctan (\frac{1}{20}) + kπ, k \in Z

Evaluate

3 x - 5 y \times 12 = 0

Evaluate

3 x - 60 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 - 60 sin (θ) \times r = 0

Factor the expression

(3 cos (θ) - 60 sin (θ)) r = 0

Separate into possible cases

r = 0 3 cos (θ) - 60 sin (θ) = 0

Solution

More Steps

Evaluate

3 cos (θ) - 60 sin (θ) = 0

Move the expression to the right side

- 60 sin (θ) = 0 - 3 cos (θ)

Subtract the terms

- 60 sin (θ) = - 3 cos (θ)

Divide both sides

\frac{- 60 sin ( θ )}{cos ( θ )} = - 3

Divide the terms

More Steps

Evaluate

\frac{- 60 sin ( θ )}{cos ( θ )}

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

- \frac{60 sin ( θ )}{cos ( θ )}

Rewrite the expression

- 60 cos^{- 1} (θ) sin (θ)

Rewrite the expression

- 60 tan (θ)

- 60 tan (θ) = - 3

Multiply both sides of the equation by - \frac{1}{60}

- 60 tan (θ) (- \frac{1}{60}) = - 3 (- \frac{1}{60})

Calculate

tan (θ) = - 3 (- \frac{1}{60})

Calculate

More Steps

Evaluate

- 3 (- \frac{1}{60})

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

3 \times \frac{1}{60}

Reduce the numbers

1 \times \frac{1}{20}

Multiply the numbers

\frac{1}{20}

tan (θ) = \frac{1}{20}

Use the inverse trigonometric function

θ = arctan (\frac{1}{20})

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

θ = arctan (\frac{1}{20}) + kπ, k \in Z

r = 0 θ = arctan (\frac{1}{20}) + 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{1}{20}

Calculate

3 x - 5 y 12 = 0

Simplify the expression

3 x - 60 y = 0

Take the derivative of both sides

\frac{d}{d x} (3 x - 60 y) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 x - 60 y)

Use differentiation rules

\frac{d}{d x} (3 x) + \frac{d}{d x} (- 60 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} (- 60 y)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 60 y)

Use differentiation rules

\frac{d}{d y} (- 60 y) \times \frac{d y}{d x}

Evaluate the derivative

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

3 - 60 \frac{d y}{d x}

3 - 60 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

3 - 60 \frac{d y}{d x} = 0

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

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

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

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

Change the signs on both sides of the equation

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

Divide both sides

\frac{60 \frac{d y}{d x}}{60} = \frac{3}{60}

Divide the numbers

\frac{d y}{d x} = \frac{3}{60}

Solution

\frac{d y}{d x} = \frac{1}{20}

Show Solution

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 12 = 0

Simplify the expression

3 x - 60 y = 0

Take the derivative of both sides

\frac{d}{d x} (3 x - 60 y) = \frac{d}{d x} (0)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (3 x - 60 y)

Use differentiation rules

\frac{d}{d x} (3 x) + \frac{d}{d x} (- 60 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} (- 60 y)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (- 60 y)

Use differentiation rules

\frac{d}{d y} (- 60 y) \times \frac{d y}{d x}

Evaluate the derivative

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

3 - 60 \frac{d y}{d x}

3 - 60 \frac{d y}{d x} = \frac{d}{d x} (0)

Calculate the derivative

3 - 60 \frac{d y}{d x} = 0

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

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

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

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

Change the signs on both sides of the equation

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

Divide both sides

\frac{60 \frac{d y}{d x}}{60} = \frac{3}{60}

Divide the numbers

\frac{d y}{d x} = \frac{3}{60}

Cancel out the common factor 3

\frac{d y}{d x} = \frac{1}{20}

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\frac{1}{20})

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{1}{20})

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