Calculus Examples

2 {(x^{2} + y^{2})}^{2} = 25 x y^{2}

$2 {(x^{2} + y^{2})}^{2} = 25 x y^{2}$ ;

$(2, 1)$

Step 1

Find the first derivative and evaluate at

$x = 2$ and

$y = 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (2 {(x^{2} + y^{2})}^{2}) = \frac{d}{d x} (25 x y^{2})$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 {(x^{2} + y^{2})}^{2}$ with respect to

$x$ is

$2 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$ .

$2 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$

Step 1.2.2

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{2}$ and

$g (x) = x^{2} + y^{2}$ .

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Step 1.2.2.1

To apply the Chain Rule, set

$u_{1}$ as

$x^{2} + y^{2}$ .

$2 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [x^{2} + y^{2}])$

Step 1.2.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is

$n {u_{1}}^{n - 1}$ where

$n = 2$ .

$2 (2 u_{1} \frac{d}{d x} [x^{2} + y^{2}])$

Step 1.2.2.3

Replace all occurrences of

$u_{1}$ with

$x^{2} + y^{2}$ .

$2 (2 (x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

Step 1.2.3

Differentiate.

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Step 1.2.3.1

Multiply

$2$ by

$2$ .

$4 ((x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

Step 1.2.3.2

By the Sum Rule, the derivative of

$x^{2} + y^{2}$ with respect to

$x$ is

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]$ .

$4 (x^{2} + y^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

Step 1.2.3.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$4 (x^{2} + y^{2}) (2 x + \frac{d}{d x} [y^{2}])$

Step 1.2.4

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{2}$ and

$g (x) = y$ .

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Step 1.2.4.1

To apply the Chain Rule, set

$u_{2}$ as

$y$ .

$4 (x^{2} + y^{2}) (2 x + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

Step 1.2.4.2

Differentiate using the Power Rule which states that

$\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is

$n {u_{2}}^{n - 1}$ where

$n = 2$ .

$4 (x^{2} + y^{2}) (2 x + 2 u_{2} \frac{d}{d x} [y])$

Step 1.2.4.3

Replace all occurrences of

$u_{2}$ with

$y$ .

$4 (x^{2} + y^{2}) (2 x + 2 y \frac{d}{d x} [y])$

Step 1.2.5

Rewrite

$\frac{d}{d x} [y]$ as

$y'$ .

$4 (x^{2} + y^{2}) (2 x + 2 y y')$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Apply the distributive property.

$(4 x^{2} + 4 y^{2}) (2 x + 2 y y')$

Step 1.2.6.2

Reorder the factors of

$(4 x^{2} + 4 y^{2}) (2 x + 2 y y')$ .

$(2 x + 2 y y') (4 x^{2} + 4 y^{2})$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Since

$25$ is constant with respect to

$x$ , the derivative of

$25 x y^{2}$ with respect to

$x$ is

$25 \frac{d}{d x} [x y^{2}]$ .

$25 \frac{d}{d x} [x y^{2}]$

Step 1.3.2

Differentiate using the Product Rule which states that

$\frac{d}{d x} [f (x) g (x)]$ is

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

$f (x) = x$ and

$g (x) = y^{2}$ .

$25 (x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x])$

Step 1.3.3

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{2}$ and

$g (x) = y$ .

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Step 1.3.3.1

To apply the Chain Rule, set

$u$ as

$y$ .

$25 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = 2$ .

$25 (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.3.3

Replace all occurrences of

$u$ with

$y$ .

$25 (x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 1.3.4

Move

$2$ to the left of

$x$ .

$25 (2 \cdot x y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x])$

Step 1.3.5

Rewrite

$\frac{d}{d x} [y]$ as

$y'$ .

$25 (2 x y y' + y^{2} \frac{d}{d x} [x])$

Step 1.3.6

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$25 (2 x y y' + y^{2} \cdot 1)$

Step 1.3.7

Multiply

$y^{2}$ by

$1$ .

$25 (2 x y y' + y^{2})$

Step 1.3.8

Simplify.

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Step 1.3.8.1

Apply the distributive property.

$25 (2 x y y') + 25 y^{2}$

Step 1.3.8.2

Multiply

$2$ by

$25$ .

$50 x y y' + 25 y^{2}$

Step 1.3.8.3

Reorder terms.

$25 y^{2} + 50 x y y'$

Step 1.4

Reform the equation by setting the left side equal to the right side.

$(2 x + 2 y y') (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5

Solve for

$y'$ .

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Step 1.5.1

Simplify

$(2 x + 2 y y') (4 x^{2} + 4 y^{2})$ .

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Step 1.5.1.1

Rewrite.

$0 + 0 + (2 x + 2 y y') (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.2

Simplify by adding zeros.

$(2 x + 2 y y') (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.3

Expand

$(2 x + 2 y y') (4 x^{2} + 4 y^{2})$ using the FOIL Method.

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Step 1.5.1.3.1

Apply the distributive property.

$2 x (4 x^{2} + 4 y^{2}) + 2 y y' (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.3.2

Apply the distributive property.

$2 x (4 x^{2}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2} + 4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.3.3

Apply the distributive property.

$2 x (4 x^{2}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4

Simplify each term.

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Step 1.5.1.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 4 x \cdot x^{2} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.2

Multiply

$x$ by

$x^{2}$ by adding the exponents.

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Step 1.5.1.4.2.1

Move

$x^{2}$ .

$2 \cdot 4 (x^{2} x) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.2.2

Multiply

$x^{2}$ by

$x$ .

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Step 1.5.1.4.2.2.1

Raise

$x$ to the power of

$1$ .

$2 \cdot 4 (x^{2} x^{1}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.2.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \cdot 4 x^{2 + 1} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.2.3

Add

$2$ and

$1$ .

$2 \cdot 4 x^{3} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.3

Multiply

$2$ by

$4$ .

$8 x^{3} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.4

Rewrite using the commutative property of multiplication.

$8 x^{3} + 2 \cdot 4 x y^{2} + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.5

Multiply

$2$ by

$4$ .

$8 x^{3} + 8 x y^{2} + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.6

Multiply

$4$ by

$2$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y y' (4 y^{2}) = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.7

Multiply

$y$ by

$y^{2}$ by adding the exponents.

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Step 1.5.1.4.7.1

Move

$y^{2}$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 (y^{2} y) y' \cdot 4 = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.7.2

Multiply

$y^{2}$ by

$y$ .

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Step 1.5.1.4.7.2.1

Raise

$y$ to the power of

$1$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 (y^{2} y^{1}) y' \cdot 4 = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.7.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y^{2 + 1} y' \cdot 4 = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.7.3

Add

$2$ and

$1$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y^{3} y' \cdot 4 = 25 y^{2} + 50 x y y'$

Step 1.5.1.4.8

Multiply

$4$ by

$2$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' = 25 y^{2} + 50 x y y'$

Step 1.5.2

Subtract

$50 x y y'$ from both sides of the equation.

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' - 50 x y y' = 25 y^{2}$

Step 1.5.3

Move all terms not containing

$y'$ to the right side of the equation.

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Step 1.5.3.1

Subtract

$8 x^{3}$ from both sides of the equation.

$8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' - 50 x y y' = 25 y^{2} - 8 x^{3}$

Step 1.5.3.2

Subtract

$8 x y^{2}$ from both sides of the equation.

$8 y y' x^{2} + 8 y^{3} y' - 50 x y y' = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4

Factor

$2 y y'$ out of

$8 y y' x^{2} + 8 y^{3} y' - 50 x y y'$ .

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Step 1.5.4.1

Factor

$2 y y'$ out of

$8 y y' x^{2}$ .

$2 y y' (4 x^{2}) + 8 y^{3} y' - 50 x y y' = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4.2

Factor

$2 y y'$ out of

$8 y^{3} y'$ .

$2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) - 50 x y y' = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4.3

Factor

$2 y y'$ out of

$- 50 x y y'$ .

$2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) + 2 y y' (- 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4.4

Factor

$2 y y'$ out of

$2 y y' (4 x^{2}) + 2 y y' (4 y^{2})$ .

$2 y y' (4 x^{2} + 4 y^{2}) + 2 y y' (- 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.4.5

Factor

$2 y y'$ out of

$2 y y' (4 x^{2} + 4 y^{2}) + 2 y y' (- 25 x)$ .

$2 y y' (4 x^{2} + 4 y^{2} - 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$

Step 1.5.5

Divide each term in

$2 y y' (4 x^{2} + 4 y^{2} - 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$ by

$2 y (4 x^{2} + 4 y^{2} - 25 x)$ and simplify.

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Step 1.5.5.1

Divide each term in

$2 y y' (4 x^{2} + 4 y^{2} - 25 x) = 25 y^{2} - 8 x^{3} - 8 x y^{2}$ by

$2 y (4 x^{2} + 4 y^{2} - 25 x)$ .

$\frac{2 y y' (4 x^{2} + 4 y^{2} - 25 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)} = \frac{25 y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.2

Simplify the left side.

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Step 1.5.5.2.1

Cancel the common factor of

$2$ .

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Step 1.5.5.2.1.1

Cancel the common factor.

Step 1.5.5.2.1.2

Rewrite the expression.

$\frac{y y' (4 x^{2} + 4 y^{2} - 25 x)}{y (4 x^{2} + 4 y^{2} - 25 x)} = \frac{25 y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.2.2

Cancel the common factor of

$y$ .

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Step 1.5.5.2.2.1

Cancel the common factor.

Step 1.5.5.2.2.2

Rewrite the expression.

$\frac{y' (4 x^{2} + 4 y^{2} - 25 x)}{4 x^{2} + 4 y^{2} - 25 x} = \frac{25 y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.2.3

Cancel the common factor of

$4 x^{2} + 4 y^{2} - 25 x$ .

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Step 1.5.5.2.3.1

Cancel the common factor.

Step 1.5.5.2.3.2

Divide

$y'$ by

$1$ .

$y' = \frac{25 y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3

Simplify the right side.

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Step 1.5.5.3.1

Simplify each term.

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Step 1.5.5.3.1.1

Cancel the common factor of

$y^{2}$ and

$y$ .

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Step 1.5.5.3.1.1.1

Factor

$y$ out of

$25 y^{2}$ .

$y' = \frac{y (25 y)}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.1.2

Cancel the common factors.

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Step 1.5.5.3.1.1.2.1

Factor

$y$ out of

$2 y (4 x^{2} + 4 y^{2} - 25 x)$ .

$y' = \frac{y (25 y)}{y (2 (4 x^{2} + 4 y^{2} - 25 x))} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{y (25 y)}{y (2 (4 x^{2} + 4 y^{2} - 25 x))} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.2

Cancel the common factor of

$- 8$ and

$2$ .

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Step 1.5.5.3.1.2.1

Factor

$2$ out of

$- 8 x^{3}$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x^{3})}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.2.2

Cancel the common factors.

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Step 1.5.5.3.1.2.2.1

Factor

$2$ out of

$2 y (4 x^{2} + 4 y^{2} - 25 x)$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x^{3})}{2 (y (4 x^{2} + 4 y^{2} - 25 x))} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.2.2.2

Cancel the common factor.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x^{3})}{2 (y (4 x^{2} + 4 y^{2} - 25 x))} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.2.2.3

Rewrite the expression.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.3

Move the negative in front of the fraction.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.4

Cancel the common factor of

$- 8$ and

$2$ .

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Step 1.5.5.3.1.4.1

Factor

$2$ out of

$- 8 x y^{2}$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x y^{2})}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.4.2

Cancel the common factors.

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Step 1.5.5.3.1.4.2.1

Factor

$2$ out of

$2 y (4 x^{2} + 4 y^{2} - 25 x)$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x y^{2})}{2 (y (4 x^{2} + 4 y^{2} - 25 x))}$

Step 1.5.5.3.1.4.2.2

Cancel the common factor.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{2 (- 4 x y^{2})}{2 (y (4 x^{2} + 4 y^{2} - 25 x))}$

Step 1.5.5.3.1.4.2.3

Rewrite the expression.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 4 x y^{2}}{y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.5

Cancel the common factor of

$y^{2}$ and

$y$ .

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Step 1.5.5.3.1.5.1

Factor

$y$ out of

$- 4 x y^{2}$ .

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (- 4 x y)}{y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.5.2

Cancel the common factors.

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Step 1.5.5.3.1.5.2.1

Cancel the common factor.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (- 4 x y)}{y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.1.5.2.2

Rewrite the expression.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{- 4 x y}{4 x^{2} + 4 y^{2} - 25 x}$

Step 1.5.5.3.1.6

Move the negative in front of the fraction.

$y' = \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x y}{4 x^{2} + 4 y^{2} - 25 x}$

Step 1.5.5.3.2

To write

$- \frac{4 x y}{4 x^{2} + 4 y^{2} - 25 x}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x y}{4 x^{2} + 4 y^{2} - 25 x} \cdot \frac{2}{2}$

Step 1.5.5.3.3

Write each expression with a common denominator of

$2 (4 x^{2} + 4 y^{2} - 25 x)$ , by multiplying each by an appropriate factor of

$1$ .

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Step 1.5.5.3.3.1

Multiply

$\frac{4 x y}{4 x^{2} + 4 y^{2} - 25 x}$ by

$\frac{2}{2}$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x y \cdot 2}{(4 x^{2} + 4 y^{2} - 25 x) \cdot 2}$

Step 1.5.5.3.3.2

Reorder the factors of

$(4 x^{2} + 4 y^{2} - 25 x) \cdot 2$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{25 y}{2 (4 x^{2} + 4 y^{2} - 25 x)} - \frac{4 x y \cdot 2}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.4

Combine the numerators over the common denominator.

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{25 y - 4 x y \cdot 2}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.5

Simplify the numerator.

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Step 1.5.5.3.5.1

Factor

$y$ out of

$25 y - 4 x y \cdot 2$ .

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Step 1.5.5.3.5.1.1

Factor

$y$ out of

$25 y$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y \cdot 25 - 4 x y \cdot 2}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.5.1.2

Factor

$y$ out of

$- 4 x y \cdot 2$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y \cdot 25 + y (- 4 x \cdot 2)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.5.1.3

Factor

$y$ out of

$y \cdot 25 + y (- 4 x \cdot 2)$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (25 - 4 x \cdot 2)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.5.2

Multiply

$2$ by

$- 4$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.6

To write

$- \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} \cdot \frac{2}{2} + \frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.7

To write

$\frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$ as a fraction with a common denominator, multiply by

$\frac{y}{y}$ .

$y' = - \frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)} \cdot \frac{2}{2} + \frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)} \cdot \frac{y}{y}$

Step 1.5.5.3.8

Write each expression with a common denominator of

$y (4 x^{2} + 4 y^{2} - 25 x) \cdot 2$ , by multiplying each by an appropriate factor of

$1$ .

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Step 1.5.5.3.8.1

Multiply

$\frac{4 x^{3}}{y (4 x^{2} + 4 y^{2} - 25 x)}$ by

$\frac{2}{2}$ .

$y' = - \frac{4 x^{3} \cdot 2}{y (4 x^{2} + 4 y^{2} - 25 x) \cdot 2} + \frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)} \cdot \frac{y}{y}$

Step 1.5.5.3.8.2

Multiply

$\frac{y (25 - 8 x)}{2 (4 x^{2} + 4 y^{2} - 25 x)}$ by

$\frac{y}{y}$ .

$y' = - \frac{4 x^{3} \cdot 2}{y (4 x^{2} + 4 y^{2} - 25 x) \cdot 2} + \frac{y (25 - 8 x) y}{2 (4 x^{2} + 4 y^{2} - 25 x) y}$

Step 1.5.5.3.8.3

Reorder the factors of

$y (4 x^{2} + 4 y^{2} - 25 x) \cdot 2$ .

$y' = - \frac{4 x^{3} \cdot 2}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (25 - 8 x) y}{2 (4 x^{2} + 4 y^{2} - 25 x) y}$

Step 1.5.5.3.8.4

Reorder the factors of

$2 (4 x^{2} + 4 y^{2} - 25 x) y$ .

$y' = - \frac{4 x^{3} \cdot 2}{2 y (4 x^{2} + 4 y^{2} - 25 x)} + \frac{y (25 - 8 x) y}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.9

Combine the numerators over the common denominator.

$y' = \frac{- 4 x^{3} \cdot 2 + y (25 - 8 x) y}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10

Simplify the numerator.

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Step 1.5.5.3.10.1

Multiply

$2$ by

$- 4$ .

$y' = \frac{- 8 x^{3} + y (25 - 8 x) y}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.2

Multiply

$y$ by

$y$ by adding the exponents.

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Step 1.5.5.3.10.2.1

Move

$y$ .

$y' = \frac{- 8 x^{3} + y \cdot y (25 - 8 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.2.2

Multiply

$y$ by

$y$ .

$y' = \frac{- 8 x^{3} + y^{2} (25 - 8 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.3

Apply the distributive property.

$y' = \frac{- 8 x^{3} + y^{2} \cdot 25 + y^{2} (- 8 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.4

Move

$25$ to the left of

$y^{2}$ .

$y' = \frac{- 8 x^{3} + 25 \cdot y^{2} + y^{2} (- 8 x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.10.5

Rewrite using the commutative property of multiplication.

$y' = \frac{- 8 x^{3} + 25 y^{2} - 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11

Simplify with factoring out.

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Step 1.5.5.3.11.1

Factor

$- 1$ out of

$- 8 x^{3}$ .

$y' = \frac{- (8 x^{3}) + 25 y^{2} - 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.2

Factor

$- 1$ out of

$25 y^{2}$ .

$y' = \frac{- (8 x^{3}) - (- 25 y^{2}) - 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.3

Factor

$- 1$ out of

$- (8 x^{3}) - (- 25 y^{2})$ .

$y' = \frac{- (8 x^{3} - 25 y^{2}) - 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.4

Factor

$- 1$ out of

$- 8 y^{2} x$ .

$y' = \frac{- (8 x^{3} - 25 y^{2}) - (8 y^{2} x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.5

Factor

$- 1$ out of

$- (8 x^{3} - 25 y^{2}) - (8 y^{2} x)$ .

$y' = \frac{- (8 x^{3} - 25 y^{2} + 8 y^{2} x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.6

Simplify the expression.

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Step 1.5.5.3.11.6.1

Rewrite

$- (8 x^{3} - 25 y^{2} + 8 y^{2} x)$ as

$- 1 (8 x^{3} - 25 y^{2} + 8 y^{2} x)$ .

$y' = \frac{- 1 (8 x^{3} - 25 y^{2} + 8 y^{2} x)}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.5.5.3.11.6.2

Move the negative in front of the fraction.

$y' = - \frac{8 x^{3} - 25 y^{2} + 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.6

Replace

$y'$ with

$\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{8 x^{3} - 25 y^{2} + 8 y^{2} x}{2 y (4 x^{2} + 4 y^{2} - 25 x)}$

Step 1.7

Evaluate at

$x = 2$ and

$y = 1$ .

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Step 1.7.1

Replace the variable

$x$ with

$2$ in the expression.

$- \frac{8 {(2)}^{3} - 25 y^{2} + 8 y^{2} (2)}{2 y (4 {(2)}^{2} + 4 y^{2} - 25 \cdot 2)}$

Step 1.7.2

Replace the variable

$y$ with

$1$ in the expression.

$- \frac{8 {(2)}^{3} - 25 {(1)}^{2} + 8 {(1)}^{2} (2)}{2 (1) (4 {(2)}^{2} + 4 {(1)}^{2} - 25 \cdot 2)}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Raise

$2$ to the power of

$3$ .

$- \frac{8 \cdot 8 - 25 \cdot 1^{2} + 8 \cdot 1^{2} \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.2

Multiply

$8$ by

$8$ .

$- \frac{64 - 25 \cdot 1^{2} + 8 \cdot 1^{2} \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.3

One to any power is one.

$- \frac{64 - 25 \cdot 1 + 8 \cdot 1^{2} \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.4

Multiply

$- 25$ by

$1$ .

$- \frac{64 - 25 + 8 \cdot 1^{2} \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.5

One to any power is one.

$- \frac{64 - 25 + 8 \cdot 1 \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.6

Multiply

$8 \cdot 1 \cdot 2$ .

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Step 1.7.3.6.1

Multiply

$8$ by

$1$ .

$- \frac{64 - 25 + 8 \cdot 2}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.6.2

Multiply

$8$ by

$2$ .

$- \frac{64 - 25 + 16}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.7

Subtract

$25$ from

$64$ .

$- \frac{39 + 16}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.3.8

Add

$39$ and

$16$ .

$- \frac{55}{2 (1) (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.4

Multiply

$2$ by

$1$ .

$- \frac{55}{2 (4 \cdot 2^{2} + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Raise

$2$ to the power of

$2$ .

$- \frac{55}{2 (4 \cdot 4 + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.5.2

Multiply

$4$ by

$4$ .

$- \frac{55}{2 (16 + 4 \cdot 1^{2} - 25 \cdot 2)}$

Step 1.7.5.3

One to any power is one.

$- \frac{55}{2 (16 + 4 \cdot 1 - 25 \cdot 2)}$

Step 1.7.5.4

Multiply

$4$ by

$1$ .

$- \frac{55}{2 (16 + 4 - 25 \cdot 2)}$

Step 1.7.5.5

Multiply

$- 25$ by

$2$ .

$- \frac{55}{2 (16 + 4 - 50)}$

Step 1.7.5.6

Add

$16$ and

$4$ .

$- \frac{55}{2 (20 - 50)}$

Step 1.7.5.7

Subtract

$50$ from

$20$ .

$- \frac{55}{2 \cdot - 30}$

Step 1.7.6

Reduce the expression by cancelling the common factors.

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Step 1.7.6.1

Multiply

$2$ by

$- 30$ .

$- \frac{55}{- 60}$

Step 1.7.6.2

Cancel the common factor of

$55$ and

$- 60$ .

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Step 1.7.6.2.1

Factor

$5$ out of

$55$ .

$- \frac{5 (11)}{- 60}$

Step 1.7.6.2.2

Cancel the common factors.

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Step 1.7.6.2.2.1

Factor

$5$ out of

$- 60$ .

$- \frac{5 \cdot 11}{5 \cdot - 12}$

Step 1.7.6.2.2.2

Cancel the common factor.

$- \frac{5 \cdot 11}{5 \cdot - 12}$

Step 1.7.6.2.2.3

Rewrite the expression.

$- \frac{11}{- 12}$

Step 1.7.6.3

Move the negative in front of the fraction.

$- - \frac{11}{12}$

Step 1.7.7

Multiply

$- - \frac{11}{12}$ .

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Step 1.7.7.1

Multiply

$- 1$ by

$- 1$ .

$1 (\frac{11}{12})$

Step 1.7.7.2

Multiply

$\frac{11}{12}$ by

$1$ .

$\frac{11}{12}$

Step 2

Plug the slope and point values into the point-slope formula and solve for

$y$ .

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Step 2.1

Use the slope

$\frac{11}{12}$ and a given point

$(2, 1)$ to substitute for

$x_{1}$ and

$y_{1}$ in the point-slope form

$y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{11}{12} \cdot (x - (2))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 1 = \frac{11}{12} \cdot (x - 2)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$\frac{11}{12} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + \frac{11}{12} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = \frac{11}{12} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = \frac{11}{12} x + \frac{11}{12} \cdot - 2$

Step 2.3.1.4

Combine

$\frac{11}{12}$ and

$x$ .

$y - 1 = \frac{11 x}{12} + \frac{11}{12} \cdot - 2$

Step 2.3.1.5

Cancel the common factor of

$2$ .

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Step 2.3.1.5.1

Factor

$2$ out of

$12$ .

$y - 1 = \frac{11 x}{12} + \frac{11}{2 (6)} \cdot - 2$

Step 2.3.1.5.2

Factor

$2$ out of

$- 2$ .

$y - 1 = \frac{11 x}{12} + \frac{11}{2 \cdot 6} \cdot (2 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 1 = \frac{11 x}{12} + \frac{11}{2 \cdot 6} \cdot (2 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 1 = \frac{11 x}{12} + \frac{11}{6} \cdot - 1$

Step 2.3.1.6

Combine

$\frac{11}{6}$ and

$- 1$ .

$y - 1 = \frac{11 x}{12} + \frac{11 \cdot - 1}{6}$

Step 2.3.1.7

Simplify the expression.

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Step 2.3.1.7.1

Multiply

$11$ by

$- 1$ .

$y - 1 = \frac{11 x}{12} + \frac{- 11}{6}$

Step 2.3.1.7.2

Move the negative in front of the fraction.

$y - 1 = \frac{11 x}{12} - \frac{11}{6}$

Step 2.3.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 2.3.2.1

Add

$1$ to both sides of the equation.

$y = \frac{11 x}{12} - \frac{11}{6} + 1$

Step 2.3.2.2

Write

$1$ as a fraction with a common denominator.

$y = \frac{11 x}{12} - \frac{11}{6} + \frac{6}{6}$

Step 2.3.2.3

Combine the numerators over the common denominator.

$y = \frac{11 x}{12} + \frac{- 11 + 6}{6}$

Step 2.3.2.4

Add

$- 11$ and

$6$ .

$y = \frac{11 x}{12} + \frac{- 5}{6}$

Step 2.3.2.5

Move the negative in front of the fraction.

$y = \frac{11 x}{12} - \frac{5}{6}$

Step 2.3.3

Reorder terms.

$y = \frac{11}{12} x - \frac{5}{6}$

Step 3