Finite Math Examples

$3 m - n = 5$ , $2 m + 5 n = 7$

Step 1

Solve for $m$ in $3 m - n = 5$ .

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Step 1.1

Add $n$ to both sides of the equation.

$3 m = 5 + n$

$2 m + 5 n = 7$

Step 1.2

Divide each term in $3 m = 5 + n$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 m = 5 + n$ by $3$ .

$\frac{3 m}{3} = \frac{5}{3} + \frac{n}{3}$

$2 m + 5 n = 7$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 m}{3} = \frac{5}{3} + \frac{n}{3}$

$2 m + 5 n = 7$

Step 1.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{5}{3} + \frac{n}{3}$

$2 m + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$2 m + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$2 m + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$2 m + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$2 m + 5 n = 7$

Step 2

Replace all occurrences of $m$ with $\frac{5}{3} + \frac{n}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $m$ in $2 m + 5 n = 7$ with $\frac{5}{3} + \frac{n}{3}$ .

$2 (\frac{5}{3} + \frac{n}{3}) + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (\frac{5}{3} + \frac{n}{3}) + 5 n$ .

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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.

$2 (\frac{5}{3}) + 2 (\frac{n}{3}) + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.1.2

Multiply $2 (\frac{5}{3})$ .

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Step 2.2.1.1.2.1

Combine $2$ and $\frac{5}{3}$ .

$\frac{2 \cdot 5}{3} + 2 (\frac{n}{3}) + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.1.2.2

Multiply $2$ by $5$ .

$\frac{10}{3} + 2 (\frac{n}{3}) + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$\frac{10}{3} + 2 (\frac{n}{3}) + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.1.3

Combine $2$ and $\frac{n}{3}$ .

$\frac{10}{3} + \frac{2 n}{3} + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$\frac{10}{3} + \frac{2 n}{3} + 5 n = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.2

To write $5 n$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{10}{3} + \frac{2 n}{3} + 5 n \cdot \frac{3}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.3

Combine $5 n$ and $\frac{3}{3}$ .

$\frac{10}{3} + \frac{2 n}{3} + \frac{5 n \cdot 3}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{10}{3} + \frac{2 n + 5 n \cdot 3}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{10 + 2 n + 5 n \cdot 3}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.6

Multiply $3$ by $5$ .

$\frac{10 + 2 n + 15 n}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 2.2.1.7

Add $2 n$ and $15 n$ .

$\frac{10 + 17 n}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$\frac{10 + 17 n}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$\frac{10 + 17 n}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

$\frac{10 + 17 n}{3} = 7$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3

Solve for $n$ in $\frac{10 + 17 n}{3} = 7$ .

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Step 3.1

Multiply both sides by $3$ .

$\frac{10 + 17 n}{3} \cdot 3 = 7 \cdot 3$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{10 + 17 n}{3} \cdot 3$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{10 + 17 n}{3} \cdot 3 = 7 \cdot 3$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.2.1.1.1.2

Rewrite the expression.

$10 + 17 n = 7 \cdot 3$

$m = \frac{5}{3} + \frac{n}{3}$

$10 + 17 n = 7 \cdot 3$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.2.1.1.2

Reorder $10$ and $17 n$ .

$17 n + 10 = 7 \cdot 3$

$m = \frac{5}{3} + \frac{n}{3}$

$17 n + 10 = 7 \cdot 3$

$m = \frac{5}{3} + \frac{n}{3}$

$17 n + 10 = 7 \cdot 3$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $7$ by $3$ .

$17 n + 10 = 21$

$m = \frac{5}{3} + \frac{n}{3}$

$17 n + 10 = 21$

$m = \frac{5}{3} + \frac{n}{3}$

$17 n + 10 = 21$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.3

Solve for $n$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $10$ from both sides of the equation.

$17 n = 21 - 10$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.3.1.2

Subtract $10$ from $21$ .

$17 n = 11$

$m = \frac{5}{3} + \frac{n}{3}$

$17 n = 11$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.3.2

Divide each term in $17 n = 11$ by $17$ and simplify.

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Step 3.3.2.1

Divide each term in $17 n = 11$ by $17$ .

$\frac{17 n}{17} = \frac{11}{17}$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $17$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{17 n}{17} = \frac{11}{17}$

$m = \frac{5}{3} + \frac{n}{3}$

Step 3.3.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{11}{17}$

$m = \frac{5}{3} + \frac{n}{3}$

$n = \frac{11}{17}$

$m = \frac{5}{3} + \frac{n}{3}$

$n = \frac{11}{17}$

$m = \frac{5}{3} + \frac{n}{3}$

$n = \frac{11}{17}$

$m = \frac{5}{3} + \frac{n}{3}$

$n = \frac{11}{17}$

$m = \frac{5}{3} + \frac{n}{3}$

$n = \frac{11}{17}$

$m = \frac{5}{3} + \frac{n}{3}$

Step 4

Replace all occurrences of $n$ with $\frac{11}{17}$ in each equation.

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Step 4.1

Replace all occurrences of $n$ in $m = \frac{5}{3} + \frac{n}{3}$ with $\frac{11}{17}$ .

$m = \frac{5}{3} + \frac{\frac{11}{17}}{3}$

$n = \frac{11}{17}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{5}{3} + \frac{\frac{11}{17}}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$m = \frac{5 + \frac{11}{17}}{3}$

$n = \frac{11}{17}$

Step 4.2.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$m = \frac{5 \cdot \frac{17}{17} + \frac{11}{17}}{3}$

$n = \frac{11}{17}$

Step 4.2.1.3

Combine $5$ and $\frac{17}{17}$ .

$m = \frac{\frac{5 \cdot 17}{17} + \frac{11}{17}}{3}$

$n = \frac{11}{17}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$m = \frac{\frac{5 \cdot 17 + 11}{17}}{3}$

$n = \frac{11}{17}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $5$ by $17$ .

$m = \frac{\frac{85 + 11}{17}}{3}$

$n = \frac{11}{17}$

Step 4.2.1.5.2

Add $85$ and $11$ .

$m = \frac{\frac{96}{17}}{3}$

$n = \frac{11}{17}$

$m = \frac{\frac{96}{17}}{3}$

$n = \frac{11}{17}$

Step 4.2.1.6

Multiply the numerator by the reciprocal of the denominator.

$m = \frac{96}{17} \cdot \frac{1}{3}$

$n = \frac{11}{17}$

Step 4.2.1.7

Cancel the common factor of $3$ .

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Step 4.2.1.7.1

Factor $3$ out of $96$ .

$m = \frac{3 (32)}{17} \cdot \frac{1}{3}$

$n = \frac{11}{17}$

Step 4.2.1.7.2

Cancel the common factor.

$m = \frac{3 \cdot 32}{17} \cdot \frac{1}{3}$

$n = \frac{11}{17}$

Step 4.2.1.7.3

Rewrite the expression.

$m = \frac{32}{17}$

$n = \frac{11}{17}$

$m = \frac{32}{17}$

$n = \frac{11}{17}$

$m = \frac{32}{17}$

$n = \frac{11}{17}$

$m = \frac{32}{17}$

$n = \frac{11}{17}$

$m = \frac{32}{17}$

$n = \frac{11}{17}$

Step 5

List all of the solutions.

$m = \frac{32}{17}, n = \frac{11}{17}$