Solve x - y = 15 x

Question

Function

Find the x -intercept/zero

Find the y-intercept

Find the slope

x = 0

Evaluate

x - y = 15 x

To find the x -intercept,set y =0

x - 0 = 15 x

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

x = 15 x

Add or subtract both sides

x - 15 x = 0

Subtract the terms

More Steps

Evaluate

x - 15 x

Collect like terms by calculating the sum or difference of their coefficients

(1 - 15) x

Subtract the numbers

- 14 x

- 14 x = 0

Change the signs on both sides of the equation

14 x = 0

Solution

x = 0

Show Solution

Solve the equation

Solve for x

Solve for y

x = - \frac{y}{14}

Evaluate

x - y = 15 x

Move the variable to the left side

x - y - 15 x = 0

Subtract the terms

More Steps

Evaluate

x - 15 x

Collect like terms by calculating the sum or difference of their coefficients

(1 - 15) x

Subtract the numbers

- 14 x

- 14 x - y = 0

Move the constant to the right side

- 14 x = 0 + y

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

- 14 x = y

Change the signs on both sides of the equation

14 x = - y

Divide both sides

\frac{14 x}{14} = \frac{- y}{14}

Divide the numbers

x = \frac{- y}{14}

Solution

x = - \frac{y}{14}

Show Solution

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

x - y = 15 x

To test if the graph of x - y = 15 x is symmetry with respect to the origin,substitute -x for x and -y for y

- x - (- y) = 15 (- x)

Evaluate

- x + y = 15 (- x)

Evaluate

- x + y = - 15 x

Solution

Symmetry with respect to the origin

Show Solution

Rewrite the equation

Rewrite in polar form

Rewrite in standard form

Rewrite in slope-intercept form

r = 0 θ = arctan (- 14) + kπ, k \in Z

Evaluate

x - y = 15 x

Move the expression to the left side

- 14 x - y = 0

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

- 14 cos (θ) \times r - sin (θ) \times r = 0

Factor the expression

(- 14 cos (θ) - sin (θ)) r = 0

Separate into possible cases

r = 0 - 14 cos (θ) - sin (θ) = 0

Solution

More Steps

Evaluate

- 14 cos (θ) - sin (θ) = 0

Move the expression to the right side

- sin (θ) = 0 - (- 14 cos (θ))

Subtract the terms

- sin (θ) = 14 cos (θ)

Divide both sides

\frac{- sin ( θ )}{cos ( θ )} = 14

Divide the terms

More Steps

Evaluate

\frac{- sin ( θ )}{cos ( θ )}

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

- \frac{sin ( θ )}{cos ( θ )}

Rewrite the expression

- cos^{- 1} (θ) sin (θ)

Rewrite the expression

- tan (θ)

- tan (θ) = 14

Multiply both sides of the equation by - 1

- tan (θ) (- 1) = 14 (- 1)

Calculate

tan (θ) = 14 (- 1)

Calculate

tan (θ) = - 14

Use the inverse trigonometric function

θ = arctan (- 14)

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

θ = arctan (- 14) + kπ, k \in Z

r = 0 θ = arctan (- 14) + kπ, k \in Z

Show Solution

Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

\frac{d y}{d x} = - 14

Calculate

x - y = 15 x

Take the derivative of both sides

\frac{d}{d x} (x - y) = \frac{d}{d x} (15 x)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

1 + \frac{d}{d x} (- y)

Evaluate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

- \frac{d y}{d x}

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

1 - \frac{d y}{d x} = \frac{d}{d x} (15 x)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (15 x)

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

15 \times \frac{d}{d x} (x)

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

15 \times 1

Any expression multiplied by 1 remains the same

15

1 - \frac{d y}{d x} = 15

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

- \frac{d y}{d x} = 15 - 1

Subtract the numbers

- \frac{d y}{d x} = 14

Solution

\frac{d y}{d x} = - 14

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

x - y = 15 x

Take the derivative of both sides

\frac{d}{d x} (x - y) = \frac{d}{d x} (15 x)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

1 + \frac{d}{d x} (- y)

Evaluate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

- \frac{d y}{d x}

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

1 - \frac{d y}{d x} = \frac{d}{d x} (15 x)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (15 x)

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

15 \times \frac{d}{d x} (x)

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

15 \times 1

Any expression multiplied by 1 remains the same

15

1 - \frac{d y}{d x} = 15

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

- \frac{d y}{d x} = 15 - 1

Subtract the numbers

- \frac{d y}{d x} = 14

Change the signs on both sides of the equation

\frac{d y}{d x} = - 14

Take the derivative of both sides

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (- 14)

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- 14)

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