Solve 2x+y=4x-3y - ScanMath

Question

Function

Find the x -intercept/zero

Find the y-intercept

Find the slope

x = 0

Evaluate

2 x + y = 4 x - 3 y

To find the x -intercept,set y =0

2 x + 0 = 4 x - 3 \times 0

Any expression multiplied by 0 equals 0

2 x + 0 = 4 x - 0

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

2 x = 4 x - 0

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

2 x = 4 x

Add or subtract both sides

2 x - 4 x = 0

Subtract the terms

More Steps

Evaluate

2 x - 4 x

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

(2 - 4) x

Subtract the numbers

- 2 x

- 2 x = 0

Change the signs on both sides of the equation

2 x = 0

Solution

x = 0

Show Solution

Solve the equation

Solve for x

Solve for y

x = 2 y

Evaluate

2 x + y = 4 x - 3 y

Move the expression to the left side

2 x + y - 4 x = - 3 y

Move the expression to the right side

2 x - 4 x = - 3 y - y

Add and subtract

More Steps

Evaluate

2 x - 4 x

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

(2 - 4) x

Subtract the numbers

- 2 x

- 2 x = - 3 y - y

Add and subtract

More Steps

Evaluate

- 3 y - y

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

(- 3 - 1) y

Subtract the numbers

- 4 y

- 2 x = - 4 y

Change the signs on both sides of the equation

2 x = 4 y

Divide both sides

\frac{2 x}{2} = \frac{4 y}{2}

Divide the numbers

x = \frac{4 y}{2}

Solution

More Steps

Evaluate

\frac{4 y}{2}

Reduce the numbers

\frac{2 y}{1}

Calculate

2 y

x = 2 y

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

2 x + y = 4 x - 3 y

To test if the graph of 2 x + y = 4 x - 3 y is symmetry with respect to the origin,substitute -x for x and -y for y

2 (- x) - y = 4 (- x) - 3 (- y)

Evaluate

- 2 x - y = 4 (- x) - 3 (- y)

Evaluate

More Steps

Evaluate

4 (- x) - 3 (- y)

Multiply the numbers

- 4 x - 3 (- y)

Multiply the numbers

- 4 x - (- 3 y)

Rewrite the expression

- 4 x + 3 y

- 2 x - y = - 4 x + 3 y

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 (\frac{1}{2}) + kπ, k \in Z

Evaluate

2 x + y = 4 x - 3 y

Move the expression to the left side

- 2 x + 4 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 + 4 sin (θ) \times r = 0

Factor the expression

(- 2 cos (θ) + 4 sin (θ)) r = 0

Separate into possible cases

r = 0 - 2 cos (θ) + 4 sin (θ) = 0

Solution

More Steps

Evaluate

- 2 cos (θ) + 4 sin (θ) = 0

Move the expression to the right side

4 sin (θ) = 0 - (- 2 cos (θ))

Subtract the terms

4 sin (θ) = 2 cos (θ)

Divide both sides

\frac{4 sin ( θ )}{cos ( θ )} = 2

Divide the terms

More Steps

Evaluate

\frac{4 sin ( θ )}{cos ( θ )}

Rewrite the expression

4 cos^{- 1} (θ) sin (θ)

Rewrite the expression

4 tan (θ)

4 tan (θ) = 2

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

4 tan (θ) \times \frac{1}{4} = 2 \times \frac{1}{4}

Calculate

tan (θ) = 2 \times \frac{1}{4}

Calculate

More Steps

Evaluate

2 \times \frac{1}{4}

Reduce the numbers

1 \times \frac{1}{2}

Multiply the numbers

\frac{1}{2}

tan (θ) = \frac{1}{2}

Use the inverse trigonometric function

θ = arctan (\frac{1}{2})

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

θ = arctan (\frac{1}{2}) + kπ, k \in Z

r = 0 θ = arctan (\frac{1}{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} = \frac{1}{2}

Calculate

2 x + y = 4 x - 3 y

Take the derivative of both sides

\frac{d}{d x} (2 x + y) = \frac{d}{d x} (4 x - 3 y)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

More Steps

Evaluate

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

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

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

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

2 \times 1

Any expression multiplied by 1 remains the same

2

2 + \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}

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

\frac{d y}{d x}

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

2 + \frac{d y}{d x} = \frac{d}{d x} (4 x - 3 y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (4 x - 3 y)

Use differentiation rules

\frac{d}{d x} (4 x) + \frac{d}{d x} (- 3 y)

Evaluate the derivative

More Steps

Evaluate

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

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

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

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

4 \times 1

Any expression multiplied by 1 remains the same

4

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

Evaluate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

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

4 - 3 \frac{d y}{d x}

2 + \frac{d y}{d x} = 4 - 3 \frac{d y}{d x}

Move the expression to the left side

2 + \frac{d y}{d x} + 3 \frac{d y}{d x} = 4

Move the expression to the right side

\frac{d y}{d x} + 3 \frac{d y}{d x} = 4 - 2

Add and subtract

More Steps

Evaluate

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

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

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

Add the numbers

4 \frac{d y}{d x}

4 \frac{d y}{d x} = 4 - 2

Add and subtract

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

Divide both sides

\frac{4 \frac{d y}{d x}}{4} = \frac{2}{4}

Divide the numbers

\frac{d y}{d x} = \frac{2}{4}

Solution

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

2 x + y = 4 x - 3 y

Take the derivative of both sides

\frac{d}{d x} (2 x + y) = \frac{d}{d x} (4 x - 3 y)

Calculate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

More Steps

Evaluate

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

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

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

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

2 \times 1

Any expression multiplied by 1 remains the same

2

2 + \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}

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

\frac{d y}{d x}

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

2 + \frac{d y}{d x} = \frac{d}{d x} (4 x - 3 y)

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (4 x - 3 y)

Use differentiation rules

\frac{d}{d x} (4 x) + \frac{d}{d x} (- 3 y)

Evaluate the derivative

More Steps

Evaluate

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

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

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

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

4 \times 1

Any expression multiplied by 1 remains the same

4

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

Evaluate the derivative

More Steps

Evaluate

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

Use differentiation rules

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

Evaluate the derivative

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

4 - 3 \frac{d y}{d x}

2 + \frac{d y}{d x} = 4 - 3 \frac{d y}{d x}

Move the expression to the left side

2 + \frac{d y}{d x} + 3 \frac{d y}{d x} = 4

Move the expression to the right side

\frac{d y}{d x} + 3 \frac{d y}{d x} = 4 - 2

Add and subtract

More Steps

Evaluate

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

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

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

Add the numbers

4 \frac{d y}{d x}

4 \frac{d y}{d x} = 4 - 2

Add and subtract

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

Divide both sides

\frac{4 \frac{d y}{d x}}{4} = \frac{2}{4}

Divide the numbers

\frac{d y}{d x} = \frac{2}{4}

Cancel out the common factor 2

\frac{d y}{d x} = \frac{1}{2}

Take the derivative of both sides

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

Calculate the derivative

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