Finite Math Examples

$9 r - s + t = 9$ , $3 r + 2 s - 3 t = 1$ , $r - 3 s + 2 t = - 1$

Step 1

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

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Step 1.1

Subtract $9 r$ from both sides of the equation.

$- s + t = 9 - 9 r$

$3 r + 2 s - 3 t = 1$

$r - 3 s + 2 t = - 1$

Step 1.2

Add $s$ to both sides of the equation.

$t = 9 - 9 r + s$

$3 r + 2 s - 3 t = 1$

$r - 3 s + 2 t = - 1$

$t = 9 - 9 r + s$

$3 r + 2 s - 3 t = 1$

$r - 3 s + 2 t = - 1$

Step 2

Replace all occurrences of $t$ with $9 - 9 r + s$ in each equation.

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Step 2.1

Replace all occurrences of $t$ in $3 r + 2 s - 3 t = 1$ with $9 - 9 r + s$ .

$3 r + 2 s - 3 (9 - 9 r + s) = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 r + 2 s - 3 (9 - 9 r + s)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 r + 2 s - 3 \cdot 9 - 3 (- 9 r) - 3 s = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $- 3$ by $9$ .

$3 r + 2 s - 27 - 3 (- 9 r) - 3 s = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

Step 2.2.1.1.2.2

Multiply $- 9$ by $- 3$ .

$3 r + 2 s - 27 + 27 r - 3 s = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

$3 r + 2 s - 27 + 27 r - 3 s = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

$3 r + 2 s - 27 + 27 r - 3 s = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $3 r$ and $27 r$ .

$30 r + 2 s - 27 - 3 s = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

Step 2.2.1.2.2

Subtract $3 s$ from $2 s$ .

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r - 3 s + 2 t = - 1$

Step 2.3

Replace all occurrences of $t$ in $r - 3 s + 2 t = - 1$ with $9 - 9 r + s$ .

$r - 3 s + 2 (9 - 9 r + s) = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $r - 3 s + 2 (9 - 9 r + s)$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Apply the distributive property.

$r - 3 s + 2 \cdot 9 + 2 (- 9 r) + 2 s = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 2.4.1.1.2

Simplify.

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Step 2.4.1.1.2.1

Multiply $2$ by $9$ .

$r - 3 s + 18 + 2 (- 9 r) + 2 s = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 2.4.1.1.2.2

Multiply $- 9$ by $2$ .

$r - 3 s + 18 - 18 r + 2 s = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r - 3 s + 18 - 18 r + 2 s = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r - 3 s + 18 - 18 r + 2 s = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 2.4.1.2

Simplify by adding terms.

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Step 2.4.1.2.1

Subtract $18 r$ from $r$ .

$- 17 r - 3 s + 18 + 2 s = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 2.4.1.2.2

Add $- 3 s$ and $2 s$ .

$- 17 r - s + 18 = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$- 17 r - s + 18 = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$- 17 r - s + 18 = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$- 17 r - s + 18 = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$- 17 r - s + 18 = - 1$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 3

Solve for $r$ in $- 17 r - s + 18 = - 1$ .

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Step 3.1

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

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Step 3.1.1

Add $s$ to both sides of the equation.

$- 17 r + 18 = - 1 + s$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 3.1.2

Subtract $18$ from both sides of the equation.

$- 17 r = - 1 + s - 18$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 3.1.3

Subtract $18$ from $- 1$ .

$- 17 r = s - 19$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$- 17 r = s - 19$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 3.2

Divide each term in $- 17 r = s - 19$ by $- 17$ and simplify.

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Step 3.2.1

Divide each term in $- 17 r = s - 19$ by $- 17$ .

$\frac{- 17 r}{- 17} = \frac{s}{- 17} + \frac{- 19}{- 17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 17$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 17 r}{- 17} = \frac{s}{- 17} + \frac{- 19}{- 17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 3.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{s}{- 17} + \frac{- 19}{- 17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r = \frac{s}{- 17} + \frac{- 19}{- 17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r = \frac{s}{- 17} + \frac{- 19}{- 17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Move the negative in front of the fraction.

$r = - \frac{s}{17} + \frac{- 19}{- 17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 3.2.3.1.2

Dividing two negative values results in a positive value.

$r = - \frac{s}{17} + \frac{19}{17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r = - \frac{s}{17} + \frac{19}{17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r = - \frac{s}{17} + \frac{19}{17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r = - \frac{s}{17} + \frac{19}{17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

$r = - \frac{s}{17} + \frac{19}{17}$

$30 r - s - 27 = 1$

$t = 9 - 9 r + s$

Step 4

Replace all occurrences of $r$ with $- \frac{s}{17} + \frac{19}{17}$ in each equation.

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Step 4.1

Replace all occurrences of $r$ in $30 r - s - 27 = 1$ with $- \frac{s}{17} + \frac{19}{17}$ .

$30 (- \frac{s}{17} + \frac{19}{17}) - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $30 (- \frac{s}{17} + \frac{19}{17}) - s - 27$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$30 (- \frac{s}{17}) + 30 (\frac{19}{17}) - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.1.2

Multiply $30 (- \frac{s}{17})$ .

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Step 4.2.1.1.2.1

Multiply $- 1$ by $30$ .

$- 30 \frac{s}{17} + 30 (\frac{19}{17}) - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.1.2.2

Combine $- 30$ and $\frac{s}{17}$ .

$\frac{- 30 s}{17} + 30 (\frac{19}{17}) - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$\frac{- 30 s}{17} + 30 (\frac{19}{17}) - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.1.3

Multiply $30 (\frac{19}{17})$ .

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Step 4.2.1.1.3.1

Combine $30$ and $\frac{19}{17}$ .

$\frac{- 30 s}{17} + \frac{30 \cdot 19}{17} - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.1.3.2

Multiply $30$ by $19$ .

$\frac{- 30 s}{17} + \frac{570}{17} - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$\frac{- 30 s}{17} + \frac{570}{17} - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.1.4

Move the negative in front of the fraction.

$- \frac{30 s}{17} + \frac{570}{17} - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$- \frac{30 s}{17} + \frac{570}{17} - s - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.2

To write $- s$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$- \frac{30 s}{17} - s \cdot \frac{17}{17} + \frac{570}{17} - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.3

Combine $- s$ and $\frac{17}{17}$ .

$- \frac{30 s}{17} + \frac{- s \cdot 17}{17} + \frac{570}{17} - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.4

Combine the numerators over the common denominator.

$\frac{- 30 s - s \cdot 17}{17} + \frac{570}{17} - 27 = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.5

Combine the numerators over the common denominator.

$- 27 + \frac{- 30 s - s \cdot 17 + 570}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.6

Multiply $17$ by $- 1$ .

$- 27 + \frac{- 30 s - 17 s + 570}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.7

Subtract $17 s$ from $- 30 s$ .

$- 27 + \frac{- 47 s + 570}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.8

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$- 27 \cdot \frac{17}{17} + \frac{- 47 s + 570}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.9

Simplify terms.

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Step 4.2.1.9.1

Combine $- 27$ and $\frac{17}{17}$ .

$\frac{- 27 \cdot 17}{17} + \frac{- 47 s + 570}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.9.2

Combine the numerators over the common denominator.

$\frac{- 27 \cdot 17 - 47 s + 570}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$\frac{- 27 \cdot 17 - 47 s + 570}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.10

Simplify the numerator.

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Step 4.2.1.10.1

Multiply $- 27$ by $17$ .

$\frac{- 459 - 47 s + 570}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.10.2

Add $- 459$ and $570$ .

$\frac{- 47 s + 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$\frac{- 47 s + 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.11

Simplify with factoring out.

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Step 4.2.1.11.1

Factor $- 1$ out of $- 47 s$ .

$\frac{- (47 s) + 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.11.2

Rewrite $111$ as $- 1 (- 111)$ .

$\frac{- (47 s) - 1 \cdot - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.11.3

Factor $- 1$ out of $- (47 s) - 1 (- 111)$ .

$\frac{- (47 s - 111)}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.11.4

Simplify the expression.

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Step 4.2.1.11.4.1

Rewrite $- (47 s - 111)$ as $- 1 (47 s - 111)$ .

$\frac{- 1 (47 s - 111)}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.2.1.11.4.2

Move the negative in front of the fraction.

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 - 9 r + s$

Step 4.3

Replace all occurrences of $r$ in $t = 9 - 9 r + s$ with $- \frac{s}{17} + \frac{19}{17}$ .

$t = 9 - 9 (- \frac{s}{17} + \frac{19}{17}) + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $9 - 9 (- \frac{s}{17} + \frac{19}{17}) + s$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$t = 9 - 9 (- \frac{s}{17}) - 9 (\frac{19}{17}) + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.1.2

Multiply $- 9 (- \frac{s}{17})$ .

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Step 4.4.1.1.2.1

Multiply $- 1$ by $- 9$ .

$t = 9 + 9 (\frac{s}{17}) - 9 (\frac{19}{17}) + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.1.2.2

Combine $9$ and $\frac{s}{17}$ .

$t = 9 + \frac{9 s}{17} - 9 (\frac{19}{17}) + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 + \frac{9 s}{17} - 9 (\frac{19}{17}) + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.1.3

Multiply $- 9 (\frac{19}{17})$ .

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Step 4.4.1.1.3.1

Combine $- 9$ and $\frac{19}{17}$ .

$t = 9 + \frac{9 s}{17} + \frac{- 9 \cdot 19}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.1.3.2

Multiply $- 9$ by $19$ .

$t = 9 + \frac{9 s}{17} + \frac{- 171}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 + \frac{9 s}{17} + \frac{- 171}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.1.4

Move the negative in front of the fraction.

$t = 9 + \frac{9 s}{17} - \frac{171}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 9 + \frac{9 s}{17} - \frac{171}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$t = \frac{9 s}{17} + 9 \cdot \frac{17}{17} - \frac{171}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.3

Combine $9$ and $\frac{17}{17}$ .

$t = \frac{9 s}{17} + \frac{9 \cdot 17}{17} - \frac{171}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.4

Combine the numerators over the common denominator.

$t = \frac{9 s}{17} + \frac{9 \cdot 17 - 171}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.5

Simplify the numerator.

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Step 4.4.1.5.1

Multiply $9$ by $17$ .

$t = \frac{9 s}{17} + \frac{153 - 171}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.5.2

Subtract $171$ from $153$ .

$t = \frac{9 s}{17} + \frac{- 18}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = \frac{9 s}{17} + \frac{- 18}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.6

Move the negative in front of the fraction.

$t = \frac{9 s}{17} - \frac{18}{17} + s$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.7

To write $s$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$t = \frac{9 s}{17} + s \cdot \frac{17}{17} - \frac{18}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.8

Simplify terms.

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Step 4.4.1.8.1

Combine $s$ and $\frac{17}{17}$ .

$t = \frac{9 s}{17} + \frac{s \cdot 17}{17} - \frac{18}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.8.2

Combine the numerators over the common denominator.

$t = \frac{9 s + s \cdot 17}{17} - \frac{18}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.8.3

Combine the numerators over the common denominator.

$t = \frac{9 s + s \cdot 17 - 18}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = \frac{9 s + s \cdot 17 - 18}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.9

Move $17$ to the left of $s$ .

$t = \frac{9 s + 17 s - 18}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.10

Simplify terms.

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Step 4.4.1.10.1

Add $9 s$ and $17 s$ .

$t = \frac{26 s - 18}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.10.2

Factor $2$ out of $26 s - 18$ .

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Step 4.4.1.10.2.1

Factor $2$ out of $26 s$ .

$t = \frac{2 (13 s) - 18}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.10.2.2

Factor $2$ out of $- 18$ .

$t = \frac{2 (13 s) + 2 (- 9)}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 4.4.1.10.2.3

Factor $2$ out of $2 (13 s) + 2 (- 9)$ .

$t = \frac{2 (13 s - 9)}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = \frac{2 (13 s - 9)}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = \frac{2 (13 s - 9)}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = \frac{2 (13 s - 9)}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = \frac{2 (13 s - 9)}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = \frac{2 (13 s - 9)}{17}$

$- \frac{47 s - 111}{17} = 1$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5

Solve for $s$ in $- \frac{47 s - 111}{17} = 1$ .

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Step 5.1

Multiply both sides of the equation by $- 17$ .

$- 17 (- \frac{47 s - 111}{17}) = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $- 17 (- \frac{47 s - 111}{17})$ .

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Step 5.2.1.1.1

Cancel the common factor of $17$ .

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Step 5.2.1.1.1.1

Move the leading negative in $- \frac{47 s - 111}{17}$ into the numerator.

$- 17 \frac{- (47 s - 111)}{17} = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.2.1.1.1.2

Factor $17$ out of $- 17$ .

$17 (- 1) (\frac{- (47 s - 111)}{17}) = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.2.1.1.1.3

Cancel the common factor.

$17 \cdot (- 1 \frac{- (47 s - 111)}{17}) = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.2.1.1.1.4

Rewrite the expression.

$47 s - 111 = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$47 s - 111 = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.2.1.1.2

Multiply.

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Step 5.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (47 s - 111) = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.2.1.1.2.2

Multiply $47 s - 111$ by $1$ .

$47 s - 111 = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$47 s - 111 = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$47 s - 111 = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$47 s - 111 = - 17 \cdot 1$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $- 17$ by $1$ .

$47 s - 111 = - 17$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$47 s - 111 = - 17$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$47 s - 111 = - 17$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.3

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

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Step 5.3.1

Add $111$ to both sides of the equation.

$47 s = - 17 + 111$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.3.2

Add $- 17$ and $111$ .

$47 s = 94$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$47 s = 94$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.4

Divide each term in $47 s = 94$ by $47$ and simplify.

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Step 5.4.1

Divide each term in $47 s = 94$ by $47$ .

$\frac{47 s}{47} = \frac{94}{47}$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $47$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{47 s}{47} = \frac{94}{47}$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.4.2.1.2

Divide $s$ by $1$ .

$s = \frac{94}{47}$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$s = \frac{94}{47}$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$s = \frac{94}{47}$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $94$ by $47$ .

$s = 2$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$s = 2$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$s = 2$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

$s = 2$

$t = \frac{2 (13 s - 9)}{17}$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 6

Replace all occurrences of $s$ with $2$ in each equation.

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Step 6.1

Replace all occurrences of $s$ in $t = \frac{2 (13 s - 9)}{17}$ with $2$ .

$t = \frac{2 (13 (2) - 9)}{17}$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{2 (13 (2) - 9)}{17}$ .

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Step 6.2.1.1

Simplify the numerator.

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Step 6.2.1.1.1

Multiply $13$ by $2$ .

$t = \frac{2 (26 - 9)}{17}$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 6.2.1.1.2

Subtract $9$ from $26$ .

$t = \frac{2 \cdot 17}{17}$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = \frac{2 \cdot 17}{17}$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 6.2.1.2

Simplify the expression.

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Step 6.2.1.2.1

Multiply $2$ by $17$ .

$t = \frac{34}{17}$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 6.2.1.2.2

Divide $34$ by $17$ .

$t = 2$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 2$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 2$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

$t = 2$

$s = 2$

$r = - \frac{s}{17} + \frac{19}{17}$

Step 6.3

Replace all occurrences of $s$ in $r = - \frac{s}{17} + \frac{19}{17}$ with $2$ .

$r = - \frac{2}{17} + \frac{19}{17}$

$t = 2$

$s = 2$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- \frac{2}{17} + \frac{19}{17}$ .

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Step 6.4.1.1

Combine the numerators over the common denominator.

$r = \frac{- 2 + 19}{17}$

$t = 2$

$s = 2$

Step 6.4.1.2

Simplify the expression.

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Step 6.4.1.2.1

Add $- 2$ and $19$ .

$r = \frac{17}{17}$

$t = 2$

$s = 2$

Step 6.4.1.2.2

Divide $17$ by $17$ .

$r = 1$

$t = 2$

$s = 2$

$r = 1$

$t = 2$

$s = 2$

$r = 1$

$t = 2$

$s = 2$

$r = 1$

$t = 2$

$s = 2$

$r = 1$

$t = 2$

$s = 2$

Step 7

List all of the solutions.

$r = 1, t = 2, s = 2$