Solve 7(x 1) = 4y - x

Question

Function

Find the x -intercept/zero

Find the y-intercept

Find the slope

x = 0

Evaluate

7 (x \times 1) = 4 y - x

To find the x -intercept,set y =0

7 (x \times 1) = 4 \times 0 - x

Any expression multiplied by 0 equals 0

7 (x \times 1) = 0 - x

Remove the parentheses

7 x \times 1 = 0 - x

Multiply the terms

7 x = 0 - x

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

7 x = - x

Add or subtract both sides

7 x - (- x) = 0

Subtract the terms

More Steps

Evaluate

7 x - (- x)

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

(7 - (- 1)) x

Subtract the terms

More Steps

Evaluate

7 - (- 1)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

7 + 1

Add the numbers

8

8 x

8 x = 0

Solution

x = 0

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{y}{2}

Evaluate

7 (x \times 1) = 4 y - x

Remove the parentheses

7 x \times 1 = 4 y - x

Multiply the terms

7 x = 4 y - x

Move the variable to the left side

7 x + x = 4 y

Add the terms

More Steps

Evaluate

7 x + x

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

(7 + 1) x

Add the numbers

8 x

8 x = 4 y

Divide both sides

\frac{8 x}{8} = \frac{4 y}{8}

Divide the numbers

x = \frac{4 y}{8}

Solution

x = \frac{y}{2}

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

7 (x 1) = 4 y - x

Simplify the expression

7 x = 4 y - x

To test if the graph of 7 x = 4 y - x is symmetry with respect to the origin,substitute -x for x and -y for y

7 (- x) = 4 (- y) - (- x)

Evaluate

- 7 x = 4 (- y) - (- x)

Evaluate

More Steps

Evaluate

4 (- y) - (- x)

Multiply the numbers

- 4 y - (- x)

Rewrite the expression

- 4 y + x

- 7 x = - 4 y + 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 (2) + kπ, k \in Z

Evaluate

7 (x \times 1) = 4 y - x

Evaluate

More Steps

Evaluate

7 (x \times 1)

Remove the parentheses

7 x \times 1

Multiply the terms

7 x

7 x = 4 y - x

Move the expression to the left side

8 x - 4 y = 0

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

8 cos (θ) \times r - 4 sin (θ) \times r = 0

Factor the expression

(8 cos (θ) - 4 sin (θ)) r = 0

Separate into possible cases

r = 0 8 cos (θ) - 4 sin (θ) = 0

Solution

More Steps

Evaluate

8 cos (θ) - 4 sin (θ) = 0

Move the expression to the right side

- 4 sin (θ) = 0 - 8 cos (θ)

Subtract the terms

- 4 sin (θ) = - 8 cos (θ)

Divide both sides

\frac{- 4 sin ( θ )}{cos ( θ )} = - 8

Divide the terms

More Steps

Evaluate

\frac{- 4 sin ( θ )}{cos ( θ )}

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

- \frac{4 sin ( θ )}{cos ( θ )}

Rewrite the expression

- 4 cos^{- 1} (θ) sin (θ)

Rewrite the expression

- 4 tan (θ)

- 4 tan (θ) = - 8

Multiply both sides of the equation by - \frac{1}{4}

- 4 tan (θ) (- \frac{1}{4}) = - 8 (- \frac{1}{4})

Calculate

tan (θ) = - 8 (- \frac{1}{4})

Calculate

More Steps

Evaluate

- 8 (- \frac{1}{4})

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

8 \times \frac{1}{4}

Reduce the numbers

2 \times 1

Simplify

2

tan (θ) = 2

Use the inverse trigonometric function

θ = arctan (2)

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

θ = arctan (2) + kπ, k \in Z

r = 0 θ = arctan (2) + 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} = 2

Calculate

7 (x 1) = 4 y - x

Simplify the expression

7 x = 4 y - x

Take the derivative of both sides

\frac{d}{d x} (7 x) = \frac{d}{d x} (4 y - x)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (7 x)

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

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

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

7 \times 1

Any expression multiplied by 1 remains the same

7

7 = \frac{d}{d x} (4 y - x)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (4 y - x)

Use differentiation rules

\frac{d}{d x} (4 y) + \frac{d}{d x} (- x)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (4 y)

Use differentiation rules

\frac{d}{d y} (4 y) \times \frac{d y}{d x}

Evaluate the derivative

4 \frac{d y}{d x}

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

Evaluate the derivative

More Steps

Evaluate

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

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

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

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

- 1

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

7 = 4 \frac{d y}{d x} - 1

Swap the sides of the equation

4 \frac{d y}{d x} - 1 = 7

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

4 \frac{d y}{d x} = 7 + 1

Add the numbers

4 \frac{d y}{d x} = 8

Divide both sides

\frac{4 \frac{d y}{d x}}{4} = \frac{8}{4}

Divide the numbers

\frac{d y}{d x} = \frac{8}{4}

Solution

More Steps

Evaluate

\frac{8}{4}

Reduce the numbers

\frac{2}{1}

Calculate

2

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

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

7 (x 1) = 4 y - x

Simplify the expression

7 x = 4 y - x

Take the derivative of both sides

\frac{d}{d x} (7 x) = \frac{d}{d x} (4 y - x)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (7 x)

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

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

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

7 \times 1

Any expression multiplied by 1 remains the same

7

7 = \frac{d}{d x} (4 y - x)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (4 y - x)

Use differentiation rules

\frac{d}{d x} (4 y) + \frac{d}{d x} (- x)

Evaluate the derivative

More Steps

Evaluate

\frac{d}{d x} (4 y)

Use differentiation rules

\frac{d}{d y} (4 y) \times \frac{d y}{d x}

Evaluate the derivative

4 \frac{d y}{d x}

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

Evaluate the derivative

More Steps

Evaluate

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

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

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

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

- 1

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

7 = 4 \frac{d y}{d x} - 1

Swap the sides of the equation

4 \frac{d y}{d x} - 1 = 7

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

4 \frac{d y}{d x} = 7 + 1

Add the numbers

4 \frac{d y}{d x} = 8

Divide both sides

\frac{4 \frac{d y}{d x}}{4} = \frac{8}{4}

Divide the numbers

\frac{d y}{d x} = \frac{8}{4}

Divide the numbers

More Steps

Evaluate

\frac{8}{4}

Reduce the numbers

\frac{2}{1}

Calculate

2

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

Take the derivative of both sides

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

Calculate the derivative

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