Question
Function
Find the x-intercept/zero
Find the y-intercept
Find the slope
x=0
Evaluate
2x+y=4x−3y
To find the x-intercept,set y=0
2x+0=4x−3×0
Any expression multiplied by 0 equals 0
2x+0=4x−0
Removing 0 doesn't change the value,so remove it from the expression
2x=4x−0
Removing 0 doesn't change the value,so remove it from the expression
2x=4x
Add or subtract both sides
2x−4x=0
Subtract the terms
More Steps

Evaluate
2x−4x
Collect like terms by calculating the sum or difference of their coefficients
(2−4)x
Subtract the numbers
−2x
−2x=0
Change the signs on both sides of the equation
2x=0
Solution
x=0
Show Solution

Solve the equation
Solve for x
Solve for y
x=2y
Evaluate
2x+y=4x−3y
Move the expression to the left side
2x+y−4x=−3y
Move the expression to the right side
2x−4x=−3y−y
Add and subtract
More Steps

Evaluate
2x−4x
Collect like terms by calculating the sum or difference of their coefficients
(2−4)x
Subtract the numbers
−2x
−2x=−3y−y
Add and subtract
More Steps

Evaluate
−3y−y
Collect like terms by calculating the sum or difference of their coefficients
(−3−1)y
Subtract the numbers
−4y
−2x=−4y
Change the signs on both sides of the equation
2x=4y
Divide both sides
22x=24y
Divide the numbers
x=24y
Solution
More Steps

Evaluate
24y
Reduce the numbers
12y
Calculate
2y
x=2y
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
2x+y=4x−3y
To test if the graph of 2x+y=4x−3y is symmetry with respect to the origin,substitute -x for x and -y for y
2(−x)−y=4(−x)−3(−y)
Evaluate
−2x−y=4(−x)−3(−y)
Evaluate
More Steps

Evaluate
4(−x)−3(−y)
Multiply the numbers
−4x−3(−y)
Multiply the numbers
−4x−(−3y)
Rewrite the expression
−4x+3y
−2x−y=−4x+3y
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=0θ=arctan(21)+kπ,k∈Z
Evaluate
2x+y=4x−3y
Move the expression to the left side
−2x+4y=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
−2cos(θ)×r+4sin(θ)×r=0
Factor the expression
(−2cos(θ)+4sin(θ))r=0
Separate into possible cases
r=0−2cos(θ)+4sin(θ)=0
Solution
More Steps

Evaluate
−2cos(θ)+4sin(θ)=0
Move the expression to the right side
4sin(θ)=0−(−2cos(θ))
Subtract the terms
4sin(θ)=2cos(θ)
Divide both sides
cos(θ)4sin(θ)=2
Divide the terms
More Steps

Evaluate
cos(θ)4sin(θ)
Rewrite the expression
4cos−1(θ)sin(θ)
Rewrite the expression
4tan(θ)
4tan(θ)=2
Multiply both sides of the equation by 41
4tan(θ)×41=2×41
Calculate
tan(θ)=2×41
Calculate
More Steps

Evaluate
2×41
Reduce the numbers
1×21
Multiply the numbers
21
tan(θ)=21
Use the inverse trigonometric function
θ=arctan(21)
Add the period of kπ,k∈Z to find all solutions
θ=arctan(21)+kπ,k∈Z
r=0θ=arctan(21)+kπ,k∈Z
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=21
Calculate
2x+y=4x−3y
Take the derivative of both sides
dxd(2x+y)=dxd(4x−3y)
Calculate the derivative
More Steps

Evaluate
dxd(2x+y)
Use differentiation rules
dxd(2x)+dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(2x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
2×dxd(x)
Use dxdxn=nxn−1 to find derivative
2×1
Any expression multiplied by 1 remains the same
2
2+dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
2+dxdy
2+dxdy=dxd(4x−3y)
Calculate the derivative
More Steps

Evaluate
dxd(4x−3y)
Use differentiation rules
dxd(4x)+dxd(−3y)
Evaluate the derivative
More Steps

Evaluate
dxd(4x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
4×dxd(x)
Use dxdxn=nxn−1 to find derivative
4×1
Any expression multiplied by 1 remains the same
4
4+dxd(−3y)
Evaluate the derivative
More Steps

Evaluate
dxd(−3y)
Use differentiation rules
dyd(−3y)×dxdy
Evaluate the derivative
−3dxdy
4−3dxdy
2+dxdy=4−3dxdy
Move the expression to the left side
2+dxdy+3dxdy=4
Move the expression to the right side
dxdy+3dxdy=4−2
Add and subtract
More Steps

Evaluate
dxdy+3dxdy
Collect like terms by calculating the sum or difference of their coefficients
(1+3)dxdy
Add the numbers
4dxdy
4dxdy=4−2
Add and subtract
4dxdy=2
Divide both sides
44dxdy=42
Divide the numbers
dxdy=42
Solution
dxdy=21
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=0
Calculate
2x+y=4x−3y
Take the derivative of both sides
dxd(2x+y)=dxd(4x−3y)
Calculate the derivative
More Steps

Evaluate
dxd(2x+y)
Use differentiation rules
dxd(2x)+dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(2x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
2×dxd(x)
Use dxdxn=nxn−1 to find derivative
2×1
Any expression multiplied by 1 remains the same
2
2+dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
2+dxdy
2+dxdy=dxd(4x−3y)
Calculate the derivative
More Steps

Evaluate
dxd(4x−3y)
Use differentiation rules
dxd(4x)+dxd(−3y)
Evaluate the derivative
More Steps

Evaluate
dxd(4x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
4×dxd(x)
Use dxdxn=nxn−1 to find derivative
4×1
Any expression multiplied by 1 remains the same
4
4+dxd(−3y)
Evaluate the derivative
More Steps

Evaluate
dxd(−3y)
Use differentiation rules
dyd(−3y)×dxdy
Evaluate the derivative
−3dxdy
4−3dxdy
2+dxdy=4−3dxdy
Move the expression to the left side
2+dxdy+3dxdy=4
Move the expression to the right side
dxdy+3dxdy=4−2
Add and subtract
More Steps

Evaluate
dxdy+3dxdy
Collect like terms by calculating the sum or difference of their coefficients
(1+3)dxdy
Add the numbers
4dxdy
4dxdy=4−2
Add and subtract
4dxdy=2
Divide both sides
44dxdy=42
Divide the numbers
dxdy=42
Cancel out the common factor 2
dxdy=21
Take the derivative of both sides
dxd(dxdy)=dxd(21)
Calculate the derivative
dx2d2y=dxd(21)
Solution
dx2d2y=0
Show Solution
