Calculus Examples

$f (x) = \frac{\sqrt{x} + 1}{\sqrt{x} + 5}$ ; $x = 4$

Step 1

Find the corresponding $y$ -value to $x = 4$ .

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Step 1.1

Substitute $4$ in for $x$ .

$y = \frac{\sqrt{4} + 1}{\sqrt{4} + 5}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{\sqrt{4} + 1}{\sqrt{4} + 5}$

Step 1.2.2

Simplify $\frac{\sqrt{4} + 1}{\sqrt{4} + 5}$ .

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Step 1.2.2.1

Simplify the numerator.

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Step 1.2.2.1.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{\sqrt{2^{2}} + 1}{\sqrt{4} + 5}$

Step 1.2.2.1.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{2 + 1}{\sqrt{4} + 5}$

Step 1.2.2.1.3

Add $2$ and $1$ .

$y = \frac{3}{\sqrt{4} + 5}$

Step 1.2.2.2

Simplify the denominator.

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Step 1.2.2.2.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{3}{\sqrt{2^{2}} + 5}$

Step 1.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \frac{3}{2 + 5}$

Step 1.2.2.2.3

Add $2$ and $5$ .

$y = \frac{3}{7}$

Step 2

Find the first derivative and evaluate at $x = 4$ and $y = \frac{3}{7}$ to find the slope of the tangent line.

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Step 2.1

Apply basic rules of exponents.

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{x^{\frac{1}{2}} + 1}{\sqrt{x} + 5}]$

Step 2.1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{x^{\frac{1}{2}} + 1}{x^{\frac{1}{2}} + 5}]$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{\frac{1}{2}} + 1$ and $g (x) = x^{\frac{1}{2}} + 5$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{d}{d x} [x^{\frac{1}{2}} + 1] - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $x^{\frac{1}{2}} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [1]$ .

$\frac{(x^{\frac{1}{2}} + 5) (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.3.2

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

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.5

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.6

Combine the numerators over the common denominator.

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.7.2

Subtract $2$ from $1$ .

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.8

Combine fractions.

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Step 2.8.1

Move the negative in front of the fraction.

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.8.2

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$\frac{(x^{\frac{1}{2}} + 5) (\frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.8.3

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [1]) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.9

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(x^{\frac{1}{2}} + 5) (\frac{1}{2 x^{\frac{1}{2}}} + 0) - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.10

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) \frac{d}{d x} [x^{\frac{1}{2}} + 5]}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.11

By the Sum Rule, the derivative of $x^{\frac{1}{2}} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [5]$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.12

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

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.13

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.14

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.15

Combine the numerators over the common denominator.

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.16

Simplify the numerator.

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Step 2.16.1

Multiply $- 1$ by $2$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.16.2

Subtract $2$ from $1$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.17

Combine fractions.

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Step 2.17.1

Move the negative in front of the fraction.

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.17.2

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.17.3

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [5])}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.18

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) (\frac{1}{2 x^{\frac{1}{2}}} + 0)}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.19

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$\frac{(x^{\frac{1}{2}} + 5) \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20

Simplify.

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Step 2.20.1

Apply the distributive property.

$\frac{x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} + 5 \frac{1}{2 x^{\frac{1}{2}}} - (x^{\frac{1}{2}} + 1) \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.2

Apply the distributive property.

$\frac{x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} + 5 \frac{1}{2 x^{\frac{1}{2}}} + (- x^{\frac{1}{2}} - 1 \cdot 1) \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.3

Apply the distributive property.

$\frac{x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} + 5 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 1 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4

Simplify the numerator.

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Step 2.20.4.1

Combine the opposite terms in $x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} + 5 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 1 \frac{1}{2 x^{\frac{1}{2}}}$ .

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Step 2.20.4.1.1

Subtract $x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}$ from $x^{\frac{1}{2}} \frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{5 \frac{1}{2 x^{\frac{1}{2}}} + 0 - 1 \cdot 1 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.1.2

Add $5 \frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

$\frac{5 \frac{1}{2 x^{\frac{1}{2}}} - 1 \cdot 1 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.2

Simplify each term.

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Step 2.20.4.2.1

Combine $5$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{\frac{5}{2 x^{\frac{1}{2}}} - 1 \cdot 1 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.2.2

Multiply $- 1$ by $1$ .

$\frac{\frac{5}{2 x^{\frac{1}{2}}} - 1 \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.2.3

Rewrite $- 1 \frac{1}{2 x^{\frac{1}{2}}}$ as $- \frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{\frac{5}{2 x^{\frac{1}{2}}} - \frac{1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.3

Combine the numerators over the common denominator.

$\frac{\frac{5 - 1}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.4

Subtract $1$ from $5$ .

$\frac{\frac{4}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.5

Factor $2$ out of $4$ .

$\frac{\frac{2 \cdot 2}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.6

Cancel the common factors.

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Step 2.20.4.6.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{\frac{2 \cdot 2}{2 (x^{\frac{1}{2}})}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.6.2

Cancel the common factor.

$\frac{\frac{2 \cdot 2}{2 x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.4.6.3

Rewrite the expression.

$\frac{\frac{2}{x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.5

Combine terms.

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Step 2.20.5.1

Rewrite $\frac{\frac{2}{x^{\frac{1}{2}}}}{{(x^{\frac{1}{2}} + 5)}^{2}}$ as a product.

$\frac{2}{x^{\frac{1}{2}}} \cdot \frac{1}{{(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.20.5.2

Multiply $\frac{2}{x^{\frac{1}{2}}}$ by $\frac{1}{{(x^{\frac{1}{2}} + 5)}^{2}}$ .

$\frac{2}{x^{\frac{1}{2}} {(x^{\frac{1}{2}} + 5)}^{2}}$

Step 2.21

Evaluate the derivative at $x = 4$ .

$\frac{2}{{(4)}^{\frac{1}{2}} {({(4)}^{\frac{1}{2}} + 5)}^{2}}$

Step 2.22

Simplify.

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Step 2.22.1

Simplify the denominator.

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Step 2.22.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{2}{4^{\frac{1}{2}} {({(2^{2})}^{\frac{1}{2}} + 5)}^{2}}$

Step 2.22.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2}{4^{\frac{1}{2}} {(2^{2 (\frac{1}{2})} + 5)}^{2}}$

Step 2.22.1.3

Cancel the common factor of $2$ .

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Step 2.22.1.3.1

Cancel the common factor.

$\frac{2}{4^{\frac{1}{2}} {(2^{2 (\frac{1}{2})} + 5)}^{2}}$

Step 2.22.1.3.2

Rewrite the expression.

$\frac{2}{4^{\frac{1}{2}} {(2^{1} + 5)}^{2}}$

Step 2.22.1.4

Evaluate the exponent.

$\frac{2}{4^{\frac{1}{2}} {(2 + 5)}^{2}}$

Step 2.22.1.5

Add $2$ and $5$ .

$\frac{2}{4^{\frac{1}{2}} \cdot 7^{2}}$

Step 2.22.1.6

Rewrite $4$ as $2^{2}$ .

$\frac{2}{{(2^{2})}^{\frac{1}{2}} \cdot 7^{2}}$

Step 2.22.1.7

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2}{2^{2 (\frac{1}{2})} \cdot 7^{2}}$

Step 2.22.1.8

Cancel the common factor of $2$ .

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Step 2.22.1.8.1

Cancel the common factor.

$\frac{2}{2^{2 (\frac{1}{2})} \cdot 7^{2}}$

Step 2.22.1.8.2

Rewrite the expression.

$\frac{2}{2^{1} \cdot 7^{2}}$

Step 2.22.1.9

Evaluate the exponent.

$\frac{2}{2 \cdot 7^{2}}$

Step 2.22.1.10

Raise $7$ to the power of $2$ .

$\frac{2}{2 \cdot 49}$

Step 2.22.2

Reduce the expression by cancelling the common factors.

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Step 2.22.2.1

Multiply $2$ by $49$ .

$\frac{2}{98}$

Step 2.22.2.2

Cancel the common factor of $2$ and $98$ .

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Step 2.22.2.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{98}$

Step 2.22.2.2.2

Cancel the common factors.

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Step 2.22.2.2.2.1

Factor $2$ out of $98$ .

$\frac{2 \cdot 1}{2 \cdot 49}$

Step 2.22.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 49}$

Step 2.22.2.2.2.3

Rewrite the expression.

$\frac{1}{49}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $\frac{1}{49}$ and a given point $(4, \frac{3}{7})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{3}{7}) = \frac{1}{49} \cdot (x - (4))$

Step 3.2

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

$y - \frac{3}{7} = \frac{1}{49} \cdot (x - 4)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{1}{49} \cdot (x - 4)$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{3}{7} = 0 + 0 + \frac{1}{49} \cdot (x - 4)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{3}{7} = \frac{1}{49} \cdot (x - 4)$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{3}{7} = \frac{1}{49} x + \frac{1}{49} \cdot - 4$

Step 3.3.1.4

Combine $\frac{1}{49}$ and $x$ .

$y - \frac{3}{7} = \frac{x}{49} + \frac{1}{49} \cdot - 4$

Step 3.3.1.5

Combine $\frac{1}{49}$ and $- 4$ .

$y - \frac{3}{7} = \frac{x}{49} + \frac{- 4}{49}$

Step 3.3.1.6

Move the negative in front of the fraction.

$y - \frac{3}{7} = \frac{x}{49} - \frac{4}{49}$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $\frac{3}{7}$ to both sides of the equation.

$y = \frac{x}{49} - \frac{4}{49} + \frac{3}{7}$

Step 3.3.2.2

To write $\frac{3}{7}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = \frac{x}{49} - \frac{4}{49} + \frac{3}{7} \cdot \frac{7}{7}$

Step 3.3.2.3

Write each expression with a common denominator of $49$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.3.1

Multiply $\frac{3}{7}$ by $\frac{7}{7}$ .

$y = \frac{x}{49} - \frac{4}{49} + \frac{3 \cdot 7}{7 \cdot 7}$

Step 3.3.2.3.2

Multiply $7$ by $7$ .

$y = \frac{x}{49} - \frac{4}{49} + \frac{3 \cdot 7}{49}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{49} + \frac{- 4 + 3 \cdot 7}{49}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $3$ by $7$ .

$y = \frac{x}{49} + \frac{- 4 + 21}{49}$

Step 3.3.2.5.2

Add $- 4$ and $21$ .

$y = \frac{x}{49} + \frac{17}{49}$

Step 3.3.3

Reorder terms.

$y = \frac{1}{49} x + \frac{17}{49}$

Step 4