Question
Solve the equation
Solve for x
Solve for y
x=y425
Evaluate
x×4y×2=3400
Multiply
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Evaluate
x×4y×2
Multiply the terms
x×8y
Use the commutative property to reorder the terms
8xy
8xy=3400
Rewrite the expression
8yx=3400
Divide both sides
8y8yx=8y3400
Divide the numbers
x=8y3400
Solution
x=y425
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
x×4y×2=3400
Multiply
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Evaluate
x×4y×2
Multiply the terms
x×8y
Use the commutative property to reorder the terms
8xy
8xy=3400
To test if the graph of 8xy=3400 is symmetry with respect to the origin,substitute -x for x and -y for y
8(−x)(−y)=3400
Evaluate
8xy=3400
Solution
Symmetry with respect to the origin
Show Solution

Rewrite the equation
r=∣sin(2θ)∣534sin(2θ)r=−∣sin(2θ)∣534sin(2θ)
Evaluate
x×4y×2=3400
Evaluate
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Evaluate
x×4y×2
Multiply the terms
x×8y
Use the commutative property to reorder the terms
8xy
8xy=3400
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
8cos(θ)×rsin(θ)×r=3400
Factor the expression
8cos(θ)sin(θ)×r2=3400
Simplify the expression
4sin(2θ)×r2=3400
Divide the terms
r2=sin(2θ)850
Evaluate the power
r=±sin(2θ)850
Simplify the expression
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Evaluate
sin(2θ)850
To take a root of a fraction,take the root of the numerator and denominator separately
sin(2θ)850
Simplify the radical expression
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Evaluate
850
Write the expression as a product where the root of one of the factors can be evaluated
25×34
Write the number in exponential form with the base of 5
52×34
The root of a product is equal to the product of the roots of each factor
52×34
Reduce the index of the radical and exponent with 2
534
sin(2θ)534
Multiply by the Conjugate
sin(2θ)×sin(2θ)534×sin(2θ)
Calculate
∣sin(2θ)∣534×sin(2θ)
Calculate the product
∣sin(2θ)∣534sin(2θ)
r=±∣sin(2θ)∣534sin(2θ)
Solution
r=∣sin(2θ)∣534sin(2θ)r=−∣sin(2θ)∣534sin(2θ)
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−xy
Calculate
x⋅4y⋅2=3400
Simplify the expression
8xy=3400
Take the derivative of both sides
dxd(8xy)=dxd(3400)
Calculate the derivative
More Steps

Evaluate
dxd(8xy)
Use differentiation rules
dxd(8x)×y+8x×dxd(y)
Evaluate the derivative
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Evaluate
dxd(8x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
8×dxd(x)
Use dxdxn=nxn−1 to find derivative
8×1
Any expression multiplied by 1 remains the same
8
8y+8x×dxd(y)
Evaluate the derivative
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Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
8y+8xdxdy
8y+8xdxdy=dxd(3400)
Calculate the derivative
8y+8xdxdy=0
Move the expression to the right-hand side and change its sign
8xdxdy=0−8y
Removing 0 doesn't change the value,so remove it from the expression
8xdxdy=−8y
Divide both sides
8x8xdxdy=8x−8y
Divide the numbers
dxdy=8x−8y
Solution
More Steps

Evaluate
8x−8y
Cancel out the common factor 8
x−y
Use b−a=−ba=−ba to rewrite the fraction
−xy
dxdy=−xy
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=x22y
Calculate
x⋅4y⋅2=3400
Simplify the expression
8xy=3400
Take the derivative of both sides
dxd(8xy)=dxd(3400)
Calculate the derivative
More Steps

Evaluate
dxd(8xy)
Use differentiation rules
dxd(8x)×y+8x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(8x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
8×dxd(x)
Use dxdxn=nxn−1 to find derivative
8×1
Any expression multiplied by 1 remains the same
8
8y+8x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
8y+8xdxdy
8y+8xdxdy=dxd(3400)
Calculate the derivative
8y+8xdxdy=0
Move the expression to the right-hand side and change its sign
8xdxdy=0−8y
Removing 0 doesn't change the value,so remove it from the expression
8xdxdy=−8y
Divide both sides
8x8xdxdy=8x−8y
Divide the numbers
dxdy=8x−8y
Divide the numbers
More Steps

Evaluate
8x−8y
Cancel out the common factor 8
x−y
Use b−a=−ba=−ba to rewrite the fraction
−xy
dxdy=−xy
Take the derivative of both sides
dxd(dxdy)=dxd(−xy)
Calculate the derivative
dx2d2y=dxd(−xy)
Use differentiation rules
dx2d2y=−x2dxd(y)×x−y×dxd(x)
Calculate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
dx2d2y=−x2dxdy×x−y×dxd(x)
Use dxdxn=nxn−1 to find derivative
dx2d2y=−x2dxdy×x−y×1
Use the commutative property to reorder the terms
dx2d2y=−x2xdxdy−y×1
Any expression multiplied by 1 remains the same
dx2d2y=−x2xdxdy−y
Use equation dxdy=−xy to substitute
dx2d2y=−x2x(−xy)−y
Solution
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Calculate
−x2x(−xy)−y
Multiply the terms
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Evaluate
x(−xy)
Multiplying or dividing an odd number of negative terms equals a negative
−x×xy
Cancel out the common factor x
−1×y
Multiply the terms
−y
−x2−y−y
Subtract the terms
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Simplify
−y−y
Collect like terms by calculating the sum or difference of their coefficients
(−1−1)y
Subtract the numbers
−2y
−x2−2y
Divide the terms
−(−x22y)
Calculate
x22y
dx2d2y=x22y
Show Solution

Conic
850(x′)2−850(y′)2=1
Evaluate
x×4y×2=3400
Move the expression to the left side
x×4y×2−3400=0
Calculate
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Calculate
x×4y×2−3400
Multiply
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Evaluate
x×4y×2
Multiply the terms
x×8y
Use the commutative property to reorder the terms
8xy
8xy−3400
8xy−3400=0
The coefficients A,B and C of the general equation are A=0,B=8 and C=0
A=0B=8C=0
To find the angle of rotation θ,substitute the values of A,B and C into the formula cot(2θ)=BA−C
cot(2θ)=80−0
Calculate
cot(2θ)=0
Using the unit circle,find the smallest positive angle for which the cotangent is 0
2θ=2π
Calculate
θ=4π
To rotate the axes,use the equation of rotation and substitute 4π for θ
x=x′cos(4π)−y′sin(4π)y=x′sin(4π)+y′cos(4π)
Calculate
x=x′×22−y′sin(4π)y=x′sin(4π)+y′cos(4π)
Calculate
x=x′×22−y′×22y=x′sin(4π)+y′cos(4π)
Calculate
x=x′×22−y′×22y=x′×22+y′cos(4π)
Calculate
x=x′×22−y′×22y=x′×22+y′×22
Substitute x and y into the original equation 8xy−3400=0
8(x′×22−y′×22)(x′×22+y′×22)−3400=0
Calculate
More Steps

Calculate
8(x′×22−y′×22)(x′×22+y′×22)−3400
Use the commutative property to reorder the terms
8(22x′−y′×22)(x′×22+y′×22)−3400
Use the commutative property to reorder the terms
8(22x′−22y′)(x′×22+y′×22)−3400
Use the commutative property to reorder the terms
8(22x′−22y′)(22x′+y′×22)−3400
Use the commutative property to reorder the terms
8(22x′−22y′)(22x′+22y′)−3400
Expand the expression
More Steps

Calculate
8(22x′−22y′)(22x′+22y′)
Simplify
(42×x′−42×y′)(22x′+22y′)
Apply the distributive property
42×x′×22x′+42×x′×22y′−42×y′×22x′−42×y′×22y′
Multiply the terms
4(x′)2+42×x′×22y′−42×y′×22x′−42×y′×22y′
Multiply the numbers
4(x′)2+4x′y′−42×y′×22x′−42×y′×22y′
Multiply the numbers
4(x′)2+4x′y′−4y′x′−42×y′×22y′
Multiply the terms
4(x′)2+4x′y′−4y′x′−4(y′)2
Subtract the terms
4(x′)2+0−4(y′)2
Removing 0 doesn't change the value,so remove it from the expression
4(x′)2−4(y′)2
4(x′)2−4(y′)2−3400
4(x′)2−4(y′)2−3400=0
Move the constant to the right-hand side and change its sign
4(x′)2−4(y′)2=0−(−3400)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
4(x′)2−4(y′)2=0+3400
Removing 0 doesn't change the value,so remove it from the expression
4(x′)2−4(y′)2=3400
Multiply both sides of the equation by 34001
(4(x′)2−4(y′)2)×34001=3400×34001
Multiply the terms
More Steps

Evaluate
(4(x′)2−4(y′)2)×34001
Use the the distributive property to expand the expression
4(x′)2×34001−4(y′)2×34001
Multiply the numbers
More Steps

Evaluate
4×34001
Reduce the numbers
1×8501
Multiply the numbers
8501
8501(x′)2−4(y′)2×34001
Multiply the numbers
More Steps

Evaluate
−4×34001
Reduce the numbers
−1×8501
Multiply the numbers
−8501
8501(x′)2−8501(y′)2
8501(x′)2−8501(y′)2=3400×34001
Multiply the terms
More Steps

Evaluate
3400×34001
Reduce the numbers
1×1
Simplify
1
8501(x′)2−8501(y′)2=1
Use a=a11 to transform the expression
850(x′)2−8501(y′)2=1
Solution
850(x′)2−850(y′)2=1
Show Solution
