Calculus Examples

$y^{2} (y^{2} - 16) = x^{2} (x^{2} - 17)$ , $(0, - 4)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = - 4$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2} (y^{2} - 16)) = \frac{d}{d x} (x^{2} (x^{2} - 17))$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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 $f (x) = y^{2}$ and $g (x) = y^{2} - 16$ .

$y^{2} \frac{d}{d x} [y^{2} - 16] + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.2

By the Sum Rule, the derivative of $y^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 16]$ .

$y^{2} (\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 16]) + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.3

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 1.2.3.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$y^{2} (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- 16]) + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$y^{2} (2 u_{1} \frac{d}{d x} [y] + \frac{d}{d x} [- 16]) + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.3.3

Replace all occurrences of $u_{1}$ with $y$ .

$y^{2} (2 y \frac{d}{d x} [y] + \frac{d}{d x} [- 16]) + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.4

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$y^{2} (2 y y' + \frac{d}{d x} [- 16]) + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.5

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$y^{2} (2 y y' + 0) + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.6

Add $2 y y'$ and $0$ .

$y^{2} (2 y y') + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.7

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

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Step 1.2.7.1

Move $y$ .

$y \cdot y^{2} (2 y') + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.7.2

Multiply $y$ by $y^{2}$ .

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Step 1.2.7.2.1

Raise $y$ to the power of $1$ .

$y^{1} y^{2} (2 y') + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{1 + 2} (2 y') + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.7.3

Add $1$ and $2$ .

$y^{3} (2 y') + (y^{2} - 16) \frac{d}{d x} [y^{2}]$

Step 1.2.8

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 1.2.8.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$y^{3} (2 y') + (y^{2} - 16) (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

Step 1.2.8.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$y^{3} (2 y') + (y^{2} - 16) (2 u_{2} \frac{d}{d x} [y])$

Step 1.2.8.3

Replace all occurrences of $u_{2}$ with $y$ .

$y^{3} (2 y') + (y^{2} - 16) (2 y \frac{d}{d x} [y])$

Step 1.2.9

Move $2$ to the left of $y^{2} - 16$ .

$y^{3} (2 y') + 2 \cdot (y^{2} - 16) y \frac{d}{d x} [y]$

Step 1.2.10

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$y^{3} (2 y') + 2 (y^{2} - 16) y y'$

Step 1.2.11

Simplify.

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Step 1.2.11.1

Apply the distributive property.

$y^{3} (2 y') + (2 y^{2} + 2 \cdot - 16) y y'$

Step 1.2.11.2

Apply the distributive property.

$y^{3} (2 y') + (2 y^{2} y + 2 \cdot - 16 y) y'$

Step 1.2.11.3

Apply the distributive property.

$y^{3} (2 y') + 2 y^{2} y y' + 2 \cdot - 16 y y'$

Step 1.2.11.4

Combine terms.

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Step 1.2.11.4.1

Raise $y$ to the power of $1$ .

$y^{3} (2 y') + 2 (y^{1} y^{2}) y' + 2 \cdot - 16 y y'$

Step 1.2.11.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{3} (2 y') + 2 y^{1 + 2} y' + 2 \cdot - 16 y y'$

Step 1.2.11.4.3

Add $1$ and $2$ .

$y^{3} (2 y') + 2 y^{3} y' + 2 \cdot - 16 y y'$

Step 1.2.11.4.4

Multiply $2$ by $- 16$ .

$y^{3} (2 y') + 2 y^{3} y' - 32 y y'$

Step 1.2.11.4.5

Add $y^{3} (2 y')$ and $2 y^{3} y'$ .

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Step 1.2.11.4.5.1

Reorder $y^{3}$ and $2$ .

$2 \cdot y^{3} y' + 2 y^{3} y' - 32 y y'$

Step 1.2.11.4.5.2

Add $2 \cdot y^{3} y'$ and $2 y^{3} y'$ .

$4 y^{3} y' - 32 y y'$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

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 $f (x) = x^{2}$ and $g (x) = x^{2} - 17$ .

$x^{2} \frac{d}{d x} [x^{2} - 17] + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.2

Differentiate.

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Step 1.3.2.1

By the Sum Rule, the derivative of $x^{2} - 17$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 17]$ .

$x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 17]) + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$x^{2} (2 x + \frac{d}{d x} [- 17]) + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.2.3

Since $- 17$ is constant with respect to $x$ , the derivative of $- 17$ with respect to $x$ is $0$ .

$x^{2} (2 x + 0) + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.2.4

Add $2 x$ and $0$ .

$x^{2} (2 x) + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.3

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

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Step 1.3.3.1

Move $x$ .

$x \cdot x^{2} \cdot 2 + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.3.3.2.1

Raise $x$ to the power of $1$ .

$x^{1} x^{2} \cdot 2 + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + 2} \cdot 2 + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.3.3

Add $1$ and $2$ .

$x^{3} \cdot 2 + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.4

Move $2$ to the left of $x^{3}$ .

$2 \cdot x^{3} + (x^{2} - 17) \frac{d}{d x} [x^{2}]$

Step 1.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x^{3} + (x^{2} - 17) (2 x)$

Step 1.3.6

Move $2$ to the left of $x^{2} - 17$ .

$2 x^{3} + 2 \cdot (x^{2} - 17) x$

Step 1.3.7

Simplify.

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Step 1.3.7.1

Apply the distributive property.

$2 x^{3} + (2 x^{2} + 2 \cdot - 17) x$

Step 1.3.7.2

Apply the distributive property.

$2 x^{3} + 2 x^{2} x + 2 \cdot - 17 x$

Step 1.3.7.3

Combine terms.

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Step 1.3.7.3.1

Raise $x$ to the power of $1$ .

$2 x^{3} + 2 (x^{1} x^{2}) + 2 \cdot - 17 x$

Step 1.3.7.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{3} + 2 x^{1 + 2} + 2 \cdot - 17 x$

Step 1.3.7.3.3

Add $1$ and $2$ .

$2 x^{3} + 2 x^{3} + 2 \cdot - 17 x$

Step 1.3.7.3.4

Multiply $2$ by $- 17$ .

$2 x^{3} + 2 x^{3} - 34 x$

Step 1.3.7.3.5

Add $2 x^{3}$ and $2 x^{3}$ .

$4 x^{3} - 34 x$

Step 1.4

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

$4 y^{3} y' - 32 y y' = 4 x^{3} - 34 x$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Factor $4 y y'$ out of $4 y^{3} y' - 32 y y'$ .

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Step 1.5.1.1

Factor $4 y y'$ out of $4 y^{3} y'$ .

$4 y y' y^{2} - 32 y y' = 4 x^{3} - 34 x$

Step 1.5.1.2

Factor $4 y y'$ out of $- 32 y y'$ .

$4 y y' y^{2} + 4 y y' \cdot - 8 = 4 x^{3} - 34 x$

Step 1.5.1.3

Factor $4 y y'$ out of $4 y y' y^{2} + 4 y y' \cdot - 8$ .

$4 y y' (y^{2} - 8) = 4 x^{3} - 34 x$

Step 1.5.2

Divide each term in $4 y y' (y^{2} - 8) = 4 x^{3} - 34 x$ by $4 y (y^{2} - 8)$ and simplify.

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Step 1.5.2.1

Divide each term in $4 y y' (y^{2} - 8) = 4 x^{3} - 34 x$ by $4 y (y^{2} - 8)$ .

$\frac{4 y y' (y^{2} - 8)}{4 y (y^{2} - 8)} = \frac{4 x^{3}}{4 y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $4$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{4 y y' (y^{2} - 8)}{4 y (y^{2} - 8)} = \frac{4 x^{3}}{4 y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.2.1.2

Rewrite the expression.

$\frac{y y' (y^{2} - 8)}{y (y^{2} - 8)} = \frac{4 x^{3}}{4 y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.2.2

Cancel the common factor of $y$ .

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Step 1.5.2.2.2.1

Cancel the common factor.

$\frac{y y' (y^{2} - 8)}{y (y^{2} - 8)} = \frac{4 x^{3}}{4 y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.2.2.2

Rewrite the expression.

$\frac{y' (y^{2} - 8)}{y^{2} - 8} = \frac{4 x^{3}}{4 y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.2.3

Cancel the common factor of $y^{2} - 8$ .

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Step 1.5.2.2.3.1

Cancel the common factor.

$\frac{y' (y^{2} - 8)}{y^{2} - 8} = \frac{4 x^{3}}{4 y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{4 x^{3}}{4 y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Simplify each term.

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Step 1.5.2.3.1.1

Cancel the common factor of $4$ .

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Step 1.5.2.3.1.1.1

Cancel the common factor.

$y' = \frac{4 x^{3}}{4 y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.3.1.1.2

Rewrite the expression.

$y' = \frac{x^{3}}{y (y^{2} - 8)} + \frac{- 34 x}{4 y (y^{2} - 8)}$

Step 1.5.2.3.1.2

Cancel the common factor of $- 34$ and $4$ .

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Step 1.5.2.3.1.2.1

Factor $2$ out of $- 34 x$ .

$y' = \frac{x^{3}}{y (y^{2} - 8)} + \frac{2 (- 17 x)}{4 y (y^{2} - 8)}$

Step 1.5.2.3.1.2.2

Cancel the common factors.

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Step 1.5.2.3.1.2.2.1

Factor $2$ out of $4 y (y^{2} - 8)$ .

$y' = \frac{x^{3}}{y (y^{2} - 8)} + \frac{2 (- 17 x)}{2 (2 y (y^{2} - 8))}$

Step 1.5.2.3.1.2.2.2

Cancel the common factor.

$y' = \frac{x^{3}}{y (y^{2} - 8)} + \frac{2 (- 17 x)}{2 (2 y (y^{2} - 8))}$

Step 1.5.2.3.1.2.2.3

Rewrite the expression.

$y' = \frac{x^{3}}{y (y^{2} - 8)} + \frac{- 17 x}{2 y (y^{2} - 8)}$

Step 1.5.2.3.1.3

Move the negative in front of the fraction.

$y' = \frac{x^{3}}{y (y^{2} - 8)} - \frac{17 x}{2 y (y^{2} - 8)}$

Step 1.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{x^{3}}{y (y^{2} - 8)} - \frac{17 x}{2 y (y^{2} - 8)}$

Step 1.7

Evaluate at $x = 0$ and $y = - 4$ .

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Step 1.7.1

Replace the variable $x$ with $0$ in the expression.

$\frac{{(0)}^{3}}{y (y^{2} - 8)} - \frac{17 (0)}{2 y (y^{2} - 8)}$

Step 1.7.2

Replace the variable $y$ with $- 4$ in the expression.

$\frac{{(0)}^{3}}{(- 4) ({(- 4)}^{2} - 8)} - \frac{17 (0)}{2 (- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3

Simplify each term.

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Step 1.7.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{- 4 ({(- 4)}^{2} - 8)} - \frac{17 (0)}{2 (- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3.2

Simplify the denominator.

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Step 1.7.3.2.1

Raise $- 4$ to the power of $2$ .

$\frac{0}{- 4 (16 - 8)} - \frac{17 (0)}{2 (- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3.2.2

Subtract $8$ from $16$ .

$\frac{0}{- 4 \cdot 8} - \frac{17 (0)}{2 (- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3.3

Multiply $- 4$ by $8$ .

$\frac{0}{- 32} - \frac{17 (0)}{2 (- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3.4

Divide $0$ by $- 32$ .

$0 - \frac{17 (0)}{2 (- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3.5

Cancel the common factor of $0$ and $2$ .

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Step 1.7.3.5.1

Factor $2$ out of $17 (0)$ .

$0 - \frac{2 (17 \cdot (0))}{2 (- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3.5.2

Cancel the common factors.

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Step 1.7.3.5.2.1

Factor $2$ out of $2 (- 4) ({(- 4)}^{2} - 8)$ .

$0 - \frac{2 (17 \cdot (0))}{2 ((- 4) ({(- 4)}^{2} - 8))}$

Step 1.7.3.5.2.2

Cancel the common factor.

$0 - \frac{2 (17 \cdot (0))}{2 ((- 4) ({(- 4)}^{2} - 8))}$

Step 1.7.3.5.2.3

Rewrite the expression.

$0 - \frac{17 \cdot (0)}{(- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3.6

Cancel the common factor of $0$ and $- 4$ .

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Step 1.7.3.6.1

Factor $4$ out of $17 \cdot (0)$ .

$0 - \frac{4 (17 \cdot (0))}{(- 4) ({(- 4)}^{2} - 8)}$

Step 1.7.3.6.2

Cancel the common factors.

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Step 1.7.3.6.2.1

Factor $4$ out of $(- 4) ({(- 4)}^{2} - 8)$ .

$0 - \frac{4 (17 \cdot (0))}{4 (- ({(- 4)}^{2} - 8))}$

Step 1.7.3.6.2.2

Cancel the common factor.

$0 - \frac{4 (17 \cdot (0))}{4 (- ({(- 4)}^{2} - 8))}$

Step 1.7.3.6.2.3

Rewrite the expression.

$0 - \frac{17 \cdot (0)}{- ({(- 4)}^{2} - 8)}$

Step 1.7.3.7

Multiply $17$ by $0$ .

$0 - \frac{0}{- ({(- 4)}^{2} - 8)}$

Step 1.7.3.8

Simplify the denominator.

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Step 1.7.3.8.1

Raise $- 4$ to the power of $2$ .

$0 - \frac{0}{- (16 - 8)}$

Step 1.7.3.8.2

Subtract $8$ from $16$ .

$0 - \frac{0}{- 1 \cdot 8}$

Step 1.7.3.9

Multiply $- 1$ by $8$ .

$0 - \frac{0}{- 8}$

Step 1.7.3.10

Divide $0$ by $- 8$ .

$0 - 0$

Step 1.7.3.11

Multiply $- 1$ by $0$ .

$0 + 0$

Step 1.7.4

Add $0$ and $0$ .

$0$

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 $0$ and a given point $(0, - 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 4) = 0 \cdot (x - (0))$

Step 2.2

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

$y + 4 = 0 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $0 \cdot (x + 0)$ .

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Step 2.3.1.1

Add $x$ and $0$ .

$y + 4 = 0 \cdot x$

Step 2.3.1.2

Multiply $0$ by $x$ .

$y + 4 = 0$

Step 2.3.2

Subtract $4$ from both sides of the equation.

$y = - 4$

Step 3