Question
Solve the equation
x=−∣3y−y2∣9−3y,0<y<3x=−∣3y−y2∣9−3y,y>3x=−∣3y−y2∣9−3y,y<0x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
Evaluate
(x2y2−1)×3=x2y3
Multiply the terms
3(x2y2−1)=x2y3
Rewrite the expression
3(y2x2−1)=y3x2
Expand the expression
More Steps

Evaluate
3(y2x2−1)
Apply the distributive property
3y2x2−3×1
Any expression multiplied by 1 remains the same
3y2x2−3
3y2x2−3=y3x2
Move the expression to the left side
3y2x2−3−y3x2=0
Collect like terms by calculating the sum or difference of their coefficients
(3y2−y3)x2−3=0
Move the constant to the right side
(3y2−y3)x2=3
Divide both sides
3y2−y3(3y2−y3)x2=3y2−y33
Divide the numbers
x2=3y2−y33
Take the root of both sides of the equation and remember to use both positive and negative roots
x=±3y2−y33
Simplify the expression
More Steps

Evaluate
3y2−y33
To take a root of a fraction,take the root of the numerator and denominator separately
3y2−y33
Simplify the radical expression
More Steps

Evaluate
3y2−y3
Factor the expression
y2(3−y)
The root of a product is equal to the product of the roots of each factor
y2×3−y
Reduce the index of the radical and exponent with 2
∣y∣×3−y
∣y∣×3−y3
Multiply by the Conjugate
∣y∣×3−y×3−y3×3−y
Calculate
∣y∣∣3−y∣3×3−y
Calculate
More Steps

Evaluate
3×3−y
The product of roots with the same index is equal to the root of the product
3(3−y)
Calculate the product
9−3y
∣y∣∣3−y∣9−3y
x=±∣y∣∣3−y∣9−3y
Separate the equation into 2 possible cases
x=∣y∣∣3−y∣9−3yx=−∣y∣∣3−y∣9−3y
Multiply the terms
x=∣y(3−y)∣9−3yx=−∣y∣∣3−y∣9−3y
Multiply the terms
x=∣y(3−y)∣9−3yx=−∣y(3−y)∣9−3y
Calculate
{x=−∣y(3−y)∣9−3y0<y<3{x=−∣y(3−y)∣9−3yy>3{x=−∣y(3−y)∣9−3yy<0{x=∣y(3−y)∣9−3y0<y<3{x=∣y(3−y)∣9−3yy>3{x=∣y(3−y)∣9−3yy<0
Simplify
x=−∣y(3−y)∣9−3y,0<y<3x=−∣y(3−y)∣9−3y,y>3x=−∣y(3−y)∣9−3y,y<0x=∣3y−y2∣9−3y,y<0x=∣y(3−y)∣9−3y,0<y<3x=∣y(3−y)∣9−3y,y>3
Simplify
x=−∣y(3−y)∣9−3y,0<y<3x=−∣y(3−y)∣9−3y,y>3x=−∣y(3−y)∣9−3y,y<0x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y<0x=∣y(3−y)∣9−3y,y>3
Simplify
x=−∣y(3−y)∣9−3y,0<y<3x=−∣y(3−y)∣9−3y,y>3x=−∣y(3−y)∣9−3y,y<0x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
Simplify
x=−∣3y−y2∣9−3y,y<0x=−∣y(3−y)∣9−3y,0<y<3x=−∣y(3−y)∣9−3y,y>3x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
Simplify
x=−∣3y−y2∣9−3y,0<y<3x=−∣3y−y2∣9−3y,y<0x=−∣y(3−y)∣9−3y,y>3x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
Solution
x=−∣3y−y2∣9−3y,0<y<3x=−∣3y−y2∣9−3y,y>3x=−∣3y−y2∣9−3y,y<0x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
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
Not symmetry with respect to the origin
Evaluate
(x2y2−1)×3=x2y3
Multiply the terms
3(x2y2−1)=x2y3
To test if the graph of 3(x2y2−1)=x2y3 is symmetry with respect to the origin,substitute -x for x and -y for y
3((−x)2(−y)2−1)=(−x)2(−y)3
Evaluate
3(x2y2−1)=(−x)2(−y)3
Evaluate
More Steps

Evaluate
(−x)2(−y)3
Rewrite the expression
x2(−y3)
Use the commutative property to reorder the terms
−x2y3
3(x2y2−1)=−x2y3
Solution
Not symmetry with respect to the origin
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=6x−3yx2y2−6y
Calculate
(x2y2−1)3=x2y3
Simplify the expression
3(x2y2−1)=x2y3
Take the derivative of both sides
dxd(3(x2y2−1))=dxd(x2y3)
Calculate the derivative
More Steps

Evaluate
dxd(3(x2y2−1))
Simplify
3×dxd(x2y2−1)
Calculate
3(2xy2+2x2ydxdy)
3(2xy2+2x2ydxdy)=dxd(x2y3)
Calculate the derivative
More Steps

Evaluate
dxd(x2y3)
Use differentiation rules
dxd(x2)×y3+x2×dxd(y3)
Use dxdxn=nxn−1 to find derivative
2xy3+x2×dxd(y3)
Evaluate the derivative
More Steps

Evaluate
dxd(y3)
Use differentiation rules
dyd(y3)×dxdy
Use dxdxn=nxn−1 to find derivative
3y2dxdy
2xy3+3x2y2dxdy
3(2xy2+2x2ydxdy)=2xy3+3x2y2dxdy
Expand the expression
More Steps

Evaluate
3(2xy2+2x2ydxdy)
Apply the distributive property
3×2xy2+3×2x2ydxdy
Multiply the terms
6xy2+3×2x2ydxdy
Multiply the numbers
6xy2+6x2ydxdy
6xy2+6x2ydxdy=2xy3+3x2y2dxdy
Move the expression to the left side
6xy2+6x2ydxdy−3x2y2dxdy=2xy3
Move the expression to the right side
6x2ydxdy−3x2y2dxdy=2xy3−6xy2
Collect like terms by calculating the sum or difference of their coefficients
(6x2y−3x2y2)dxdy=2xy3−6xy2
Divide both sides
6x2y−3x2y2(6x2y−3x2y2)dxdy=6x2y−3x2y22xy3−6xy2
Divide the numbers
dxdy=6x2y−3x2y22xy3−6xy2
Solution
More Steps

Evaluate
6x2y−3x2y22xy3−6xy2
Rewrite the expression
6x2y−3x2y2xy(2y2−6y)
Rewrite the expression
xy(6x−3yx)xy(2y2−6y)
Reduce the fraction
6x−3yx2y2−6y
dxdy=6x−3yx2y2−6y
Show Solution
