Calculus Examples

$y = \frac{\sqrt{x}}{x + 9}$ , $(1, 0.1)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 0.1$ to find the slope of the tangent line.

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Step 1.1

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

$\frac{d}{d x} [\frac{x^{\frac{1}{2}}}{x + 9}]$

Step 1.2

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = x + 9$ .

$\frac{(x + 9) \frac{d}{d x} [x^{\frac{1}{2}}] - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.3

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

$\frac{(x + 9) (\frac{1}{2} x^{\frac{1}{2} - 1}) - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.4

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

$\frac{(x + 9) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.5

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

$\frac{(x + 9) (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{(x + 9) (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

$\frac{(x + 9) (\frac{1}{2} x^{\frac{1 - 2}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.7.2

Subtract $2$ from $1$ .

$\frac{(x + 9) (\frac{1}{2} x^{\frac{- 1}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

$\frac{(x + 9) (\frac{1}{2} x^{- \frac{1}{2}}) - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.8.2

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

$\frac{(x + 9) \frac{x^{- \frac{1}{2}}}{2} - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.8.3

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

$\frac{(x + 9) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.9

By the Sum Rule, the derivative of $x + 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [9]$ .

$\frac{(x + 9) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [9])}{{(x + 9)}^{2}}$

Step 1.10

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

$\frac{(x + 9) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} (1 + \frac{d}{d x} [9])}{{(x + 9)}^{2}}$

Step 1.11

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{(x + 9) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} (1 + 0)}{{(x + 9)}^{2}}$

Step 1.12

Simplify the expression.

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Step 1.12.1

Add $1$ and $0$ .

$\frac{(x + 9) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}} \cdot 1}{{(x + 9)}^{2}}$

Step 1.12.2

Multiply $- 1$ by $1$ .

$\frac{(x + 9) \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13

Simplify.

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Step 1.13.1

Apply the distributive property.

$\frac{x \frac{1}{2 x^{\frac{1}{2}}} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2

Simplify the numerator.

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Step 1.13.2.1

Simplify each term.

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Step 1.13.2.1.1

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

$\frac{\frac{x}{2 x^{\frac{1}{2}}} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2.1.2

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

$\frac{\frac{x \cdot x^{- \frac{1}{2}}}{2} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2.1.3

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 1.13.2.1.3.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 1.13.2.1.3.1.1

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

$\frac{\frac{x^{1} x^{- \frac{1}{2}}}{2} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2.1.3.1.2

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

$\frac{\frac{x^{1 - \frac{1}{2}}}{2} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2.1.3.2

Write $1$ as a fraction with a common denominator.

$\frac{\frac{x^{\frac{2}{2} - \frac{1}{2}}}{2} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2.1.3.3

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{2 - 1}{2}}}{2} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2.1.3.4

Subtract $1$ from $2$ .

$\frac{\frac{x^{\frac{1}{2}}}{2} + 9 \frac{1}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2.1.4

Combine $9$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{\frac{x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}} - x^{\frac{1}{2}}}{{(x + 9)}^{2}}$

Step 1.13.2.2

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

$\frac{\frac{x^{\frac{1}{2}}}{2} - x^{\frac{1}{2}} \cdot \frac{2}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.3

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

$\frac{\frac{x^{\frac{1}{2}}}{2} + \frac{- x^{\frac{1}{2}} \cdot 2}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.4

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1}{2}} - x^{\frac{1}{2}} \cdot 2}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.5

Simplify each term.

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Step 1.13.2.5.1

Simplify the numerator.

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Step 1.13.2.5.1.1

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

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Step 1.13.2.5.1.1.1

Move $x^{\frac{1}{2}}$ .

$\frac{\frac{x^{\frac{1}{2}} - 1 \cdot 2 x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.5.1.1.2

Multiply by $1$ .

$\frac{\frac{x^{\frac{1}{2}} \cdot 1 - 1 \cdot 2 x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.5.1.1.3

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

$\frac{\frac{x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- 1 \cdot 2)}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.5.1.1.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} \cdot 1 + x^{\frac{1}{2}} (- 1 \cdot 2)$ .

$\frac{\frac{x^{\frac{1}{2}} (1 - 1 \cdot 2)}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.5.1.2

Multiply $- 1$ by $2$ .

$\frac{\frac{x^{\frac{1}{2}} (1 - 2)}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.5.1.3

Subtract $2$ from $1$ .

$\frac{\frac{x^{\frac{1}{2}} \cdot - 1}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.5.2

Move $- 1$ to the left of $x^{\frac{1}{2}}$ .

$\frac{\frac{- 1 \cdot x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.2.5.3

Move the negative in front of the fraction.

$\frac{- \frac{x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.3

Combine terms.

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Step 1.13.3.1

Multiply $\frac{- \frac{x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$ by $\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}}$ .

$\frac{2 x^{\frac{1}{2}}}{2 x^{\frac{1}{2}}} \cdot \frac{- \frac{x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}}}{{(x + 9)}^{2}}$

Step 1.13.3.2

Combine.

$\frac{2 x^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2} + \frac{9}{2 x^{\frac{1}{2}}})}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.3

Apply the distributive property.

$\frac{2 x^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2}) + 2 x^{\frac{1}{2}} \frac{9}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.4

Cancel the common factor of $2 x^{\frac{1}{2}}$ .

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Step 1.13.3.4.1

Cancel the common factor.

$\frac{2 x^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2}) + 2 x^{\frac{1}{2}} \frac{9}{2 x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.4.2

Rewrite the expression.

$\frac{2 x^{\frac{1}{2}} (- \frac{x^{\frac{1}{2}}}{2}) + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.5

Multiply $- 1$ by $2$ .

$\frac{- 2 x^{\frac{1}{2}} \frac{x^{\frac{1}{2}}}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.6

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

$\frac{x^{\frac{1}{2}} \frac{- 2 x^{\frac{1}{2}}}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.7

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

$\frac{\frac{x^{\frac{1}{2}} (- 2 x^{\frac{1}{2}})}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.8

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 1.13.3.8.1

Move $x^{\frac{1}{2}}$ .

$\frac{\frac{x^{\frac{1}{2}} x^{\frac{1}{2}} \cdot - 2}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.8.2

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

$\frac{\frac{x^{\frac{1}{2} + \frac{1}{2}} \cdot - 2}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.8.3

Combine the numerators over the common denominator.

$\frac{\frac{x^{\frac{1 + 1}{2}} \cdot - 2}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.8.4

Add $1$ and $1$ .

$\frac{\frac{x^{\frac{2}{2}} \cdot - 2}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.8.5

Divide $2$ by $2$ .

$\frac{\frac{x^{1} \cdot - 2}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.9

Simplify $x^{1} \cdot - 2$ .

$\frac{\frac{x \cdot - 2}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.10

Move $- 2$ to the left of $x$ .

$\frac{\frac{- 2 \cdot x}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.11

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

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Step 1.13.3.11.1

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

$\frac{\frac{2 (- x)}{2} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.11.2

Cancel the common factors.

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Step 1.13.3.11.2.1

Factor $2$ out of $2$ .

$\frac{\frac{2 (- x)}{2 (1)} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.11.2.2

Cancel the common factor.

$\frac{\frac{2 (- x)}{2 \cdot 1} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.11.2.3

Rewrite the expression.

$\frac{\frac{- x}{1} + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.3.11.2.4

Divide $- x$ by $1$ .

$\frac{- x + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.4

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.5

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (x) - 1 (- 9)}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.6

Factor $- 1$ out of $- (x) - 1 (- 9)$ .

$\frac{- (x - 9)}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.7

Rewrite $- (x - 9)$ as $- 1 (x - 9)$ .

$\frac{- 1 (x - 9)}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.13.8

Move the negative in front of the fraction.

$- \frac{x - 9}{2 x^{\frac{1}{2}} {(x + 9)}^{2}}$

Step 1.14

Evaluate the derivative at $x = 1$ .

$- \frac{(1) - 9}{2 {(1)}^{\frac{1}{2}} {((1) + 9)}^{2}}$

Step 1.15

Simplify.

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Step 1.15.1

Subtract $9$ from $1$ .

$- \frac{- 8}{2 \cdot 1^{\frac{1}{2}} {(1 + 9)}^{2}}$

Step 1.15.2

Simplify the denominator.

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Step 1.15.2.1

Add $1$ and $9$ .

$- \frac{- 8}{2 \cdot 1^{\frac{1}{2}} \cdot 10^{2}}$

Step 1.15.2.2

One to any power is one.

$- \frac{- 8}{2 \cdot 1 \cdot 10^{2}}$

Step 1.15.2.3

Raise $10$ to the power of $2$ .

$- \frac{- 8}{2 \cdot 1 \cdot 100}$

Step 1.15.2.4

Combine exponents.

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Step 1.15.2.4.1

Multiply $2$ by $1$ .

$- \frac{- 8}{2 \cdot 100}$

Step 1.15.2.4.2

Multiply $2$ by $100$ .

$- \frac{- 8}{200}$

Step 1.15.3

Reduce the expression by cancelling the common factors.

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Step 1.15.3.1

Cancel the common factor of $- 8$ and $200$ .

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Step 1.15.3.1.1

Factor $8$ out of $- 8$ .

$- \frac{8 (- 1)}{200}$

Step 1.15.3.1.2

Cancel the common factors.

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Step 1.15.3.1.2.1

Factor $8$ out of $200$ .

$- \frac{8 \cdot - 1}{8 \cdot 25}$

Step 1.15.3.1.2.2

Cancel the common factor.

$- \frac{8 \cdot - 1}{8 \cdot 25}$

Step 1.15.3.1.2.3

Rewrite the expression.

$- \frac{- 1}{25}$

Step 1.15.3.2

Move the negative in front of the fraction.

$- - \frac{1}{25}$

Step 1.15.4

Multiply $- - \frac{1}{25}$ .

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Step 1.15.4.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{25})$

Step 1.15.4.2

Multiply $\frac{1}{25}$ by $1$ .

$\frac{1}{25}$

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 $\frac{1}{25}$ and a given point $(1, 0.1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0.1) = \frac{1}{25} \cdot (x - (1))$

Step 2.2

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

$y - 0.1 = \frac{1}{25} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{25} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 0.1 = 0 + 0 + \frac{1}{25} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 0.1 = \frac{1}{25} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 0.1 = \frac{1}{25} x + \frac{1}{25} \cdot - 1$

Step 2.3.1.4

Combine $\frac{1}{25}$ and $x$ .

$y - 0.1 = \frac{x}{25} + \frac{1}{25} \cdot - 1$

Step 2.3.1.5

Combine $\frac{1}{25}$ and $- 1$ .

$y - 0.1 = \frac{x}{25} + \frac{- 1}{25}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 0.1 = \frac{x}{25} - \frac{1}{25}$

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 $0.1$ to both sides of the equation.

$y = \frac{x}{25} - \frac{1}{25} + 0.1$

Step 2.3.2.2

To write $0.1$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y = \frac{x}{25} - \frac{1}{25} + 0.1 \cdot \frac{25}{25}$

Step 2.3.2.3

Combine $0.1$ and $\frac{25}{25}$ .

$y = \frac{x}{25} - \frac{1}{25} + \frac{0.1 \cdot 25}{25}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{25} + \frac{- 1 + 0.1 \cdot 25}{25}$

Step 2.3.2.5

Combine the numerators over the common denominator.

$y = \frac{x - 1 + 0.1 \cdot 25}{25}$

Step 2.3.2.6

Multiply $0.1$ by $25$ .

$y = \frac{x - 1 + 2.5}{25}$

Step 2.3.2.7

Add $- 1$ and $2.5$ .

$y = \frac{x + 1.5}{25}$

Step 2.3.2.8

Split the fraction $\frac{x + 1.5}{25}$ into two fractions.

$y = \frac{x}{25} + \frac{1.5}{25}$

Step 2.3.2.9

Divide $1.5$ by $25$ .

$y = \frac{x}{25} + 0.06$

Step 2.3.3

Reorder terms.

$y = \frac{1}{25} x + 0.06$

Step 3