Solve 2(x-y)=y - ScanMath

Question

Function

Find the x -intercept/zero

Find the y-intercept

Find the slope

x = 0

Evaluate

2 (x - y) = y

To find the x -intercept,set y =0

2 (x - 0) = 0

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

2 x = 0

Solution

x = 0

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Solve the equation

Solve for x

Solve for y

x = \frac{3 y}{2}

Evaluate

2 (x - y) = y

Divide both sides

\frac{2 ( x - y )}{2} = \frac{y}{2}

Divide the numbers

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

Move the constant to the right side

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

Solution

More Steps

Evaluate

\frac{y}{2} + y

Reduce fractions to a common denominator

\frac{y}{2} + \frac{y \times 2}{2}

Write all numerators above the common denominator

\frac{y + y \times 2}{2}

Use the commutative property to reorder the terms

\frac{y + 2 y}{2}

Add the terms

More Steps

Evaluate

y + 2 y

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

(1 + 2) y

Add the numbers

3 y

\frac{3 y}{2}

x = \frac{3 y}{2}

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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 (x - y) = y

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

2 (- x - (- y)) = - y

Evaluate

2 (- x + y) = - y

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{2}{3}) + kπ, k \in Z

Evaluate

2 (x - y) = y

Move the expression to the left side

2 x - 3 y = 0

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

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

Factor the expression

(2 cos (θ) - 3 sin (θ)) r = 0

Separate into possible cases

r = 0 2 cos (θ) - 3 sin (θ) = 0

Solution

More Steps

Evaluate

2 cos (θ) - 3 sin (θ) = 0

Move the expression to the right side

- 3 sin (θ) = 0 - 2 cos (θ)

Subtract the terms

- 3 sin (θ) = - 2 cos (θ)

Divide both sides

\frac{- 3 sin ( θ )}{cos ( θ )} = - 2

Divide the terms

More Steps

Evaluate

\frac{- 3 sin ( θ )}{cos ( θ )}

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

- \frac{3 sin ( θ )}{cos ( θ )}

Rewrite the expression

- 3 cos^{- 1} (θ) sin (θ)

Rewrite the expression

- 3 tan (θ)

- 3 tan (θ) = - 2

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

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

Calculate

tan (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}

tan (θ) = \frac{2}{3}

Use the inverse trigonometric function

θ = arctan (\frac{2}{3})

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

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

r = 0 θ = arctan (\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{2}{3}

Calculate

2 (x - y) = y

Take the derivative of both sides

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

Calculate the derivative

More Steps

Evaluate

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

Simplify

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

Rewrite the expression

2 (1 - \frac{d y}{d x})

Use the the distributive property to expand the expression

2 \times 1 + 2 (- \frac{d y}{d x})

Any expression multiplied by 1 remains the same

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

Multiply the numbers

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

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

\frac{d y}{d x}

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

Move the variable to the left side

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

Subtract the terms

More Steps

Evaluate

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

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

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

Subtract the numbers

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

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

Move the constant to the right side

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

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

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

Change the signs on both sides of the equation

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

Divide both sides

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

Solution

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

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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 (x - y) = y

Take the derivative of both sides

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

Calculate the derivative

More Steps

Evaluate

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

Simplify

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

Rewrite the expression

2 (1 - \frac{d y}{d x})

Use the the distributive property to expand the expression

2 \times 1 + 2 (- \frac{d y}{d x})

Any expression multiplied by 1 remains the same

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

Multiply the numbers

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

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

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

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

\frac{d y}{d x}

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

Move the variable to the left side

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

Subtract the terms

More Steps

Evaluate

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

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

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

Subtract the numbers

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

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

Move the constant to the right side

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

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

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

Change the signs on both sides of the equation

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

Divide both sides

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

Divide the numbers

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

Take the derivative of both sides

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

Calculate the derivative

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

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