Algebra Examples

$(- 2 \frac{1}{3}, \frac{5}{4})$ , $(- \frac{9}{5}, \frac{8}{5})$

Step 1

Convert $2 \frac{1}{3}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$(- (2 + \frac{1}{3}), \frac{5}{4}) - (- \frac{9}{5}, \frac{8}{5})$

Step 1.2

Add $2$ and $\frac{1}{3}$ .

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Step 1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(- (2 \cdot \frac{3}{3} + \frac{1}{3}), \frac{5}{4}) - (- \frac{9}{5}, \frac{8}{5})$

Step 1.2.2

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

$(- (\frac{2 \cdot 3}{3} + \frac{1}{3}), \frac{5}{4}) - (- \frac{9}{5}, \frac{8}{5})$

Step 1.2.3

Combine the numerators over the common denominator.

$(- \frac{2 \cdot 3 + 1}{3}, \frac{5}{4}) - (- \frac{9}{5}, \frac{8}{5})$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $2$ by $3$ .

$(- \frac{6 + 1}{3}, \frac{5}{4}) - (- \frac{9}{5}, \frac{8}{5})$

Step 1.2.4.2

Add $6$ and $1$ .

$(- \frac{7}{3}, \frac{5}{4}) - (- \frac{9}{5}, \frac{8}{5})$

Step 2

Use the midpoint formula to find the midpoint of the line segment.

$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

Step 3

Substitute in the values for $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

$(\frac{- \frac{7}{3} - \frac{9}{5}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4

Simplify the numerator.

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Step 4.1

To write $- \frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$(\frac{- \frac{7}{3} \cdot \frac{5}{5} - \frac{9}{5}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.2

To write $- \frac{9}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{- \frac{7}{3} \cdot \frac{5}{5} - \frac{9}{5} \cdot \frac{3}{3}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{7}{3}$ by $\frac{5}{5}$ .

$(\frac{- \frac{7 \cdot 5}{3 \cdot 5} - \frac{9}{5} \cdot \frac{3}{3}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.3.2

Multiply $3$ by $5$ .

$(\frac{- \frac{7 \cdot 5}{15} - \frac{9}{5} \cdot \frac{3}{3}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.3.3

Multiply $\frac{9}{5}$ by $\frac{3}{3}$ .

$(\frac{- \frac{7 \cdot 5}{15} - \frac{9 \cdot 3}{5 \cdot 3}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.3.4

Multiply $5$ by $3$ .

$(\frac{- \frac{7 \cdot 5}{15} - \frac{9 \cdot 3}{15}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.4

Combine the numerators over the common denominator.

$(\frac{\frac{- 7 \cdot 5 - 9 \cdot 3}{15}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 7$ by $5$ .

$(\frac{\frac{- 35 - 9 \cdot 3}{15}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.5.2

Multiply $- 9$ by $3$ .

$(\frac{\frac{- 35 - 27}{15}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.5.3

Subtract $27$ from $- 35$ .

$(\frac{\frac{- 62}{15}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 4.6

Move the negative in front of the fraction.

$(\frac{- \frac{62}{15}}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{62}{15} \cdot \frac{1}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Move the leading negative in $- \frac{62}{15}$ into the numerator.

$(\frac{- 62}{15} \cdot \frac{1}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 6.2

Factor $2$ out of $- 62$ .

$(\frac{2 (- 31)}{15} \cdot \frac{1}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 6.3

Cancel the common factor.

$(\frac{2 \cdot - 31}{15} \cdot \frac{1}{2}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 6.4

Rewrite the expression.

$(\frac{- 31}{15}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 7

Move the negative in front of the fraction.

$(- \frac{31}{15}, \frac{\frac{5}{4} + \frac{8}{5}}{2})$

Step 8

Simplify the numerator.

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Step 8.1

To write $\frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$(- \frac{31}{15}, \frac{\frac{5}{4} \cdot \frac{5}{5} + \frac{8}{5}}{2})$

Step 8.2

To write $\frac{8}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(- \frac{31}{15}, \frac{\frac{5}{4} \cdot \frac{5}{5} + \frac{8}{5} \cdot \frac{4}{4}}{2})$

Step 8.3

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.1

Multiply $\frac{5}{4}$ by $\frac{5}{5}$ .

$(- \frac{31}{15}, \frac{\frac{5 \cdot 5}{4 \cdot 5} + \frac{8}{5} \cdot \frac{4}{4}}{2})$

Step 8.3.2

Multiply $4$ by $5$ .

$(- \frac{31}{15}, \frac{\frac{5 \cdot 5}{20} + \frac{8}{5} \cdot \frac{4}{4}}{2})$

Step 8.3.3

Multiply $\frac{8}{5}$ by $\frac{4}{4}$ .

$(- \frac{31}{15}, \frac{\frac{5 \cdot 5}{20} + \frac{8 \cdot 4}{5 \cdot 4}}{2})$

Step 8.3.4

Multiply $5$ by $4$ .

$(- \frac{31}{15}, \frac{\frac{5 \cdot 5}{20} + \frac{8 \cdot 4}{20}}{2})$

Step 8.4

Combine the numerators over the common denominator.

$(- \frac{31}{15}, \frac{\frac{5 \cdot 5 + 8 \cdot 4}{20}}{2})$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $5$ by $5$ .

$(- \frac{31}{15}, \frac{\frac{25 + 8 \cdot 4}{20}}{2})$

Step 8.5.2

Multiply $8$ by $4$ .

$(- \frac{31}{15}, \frac{\frac{25 + 32}{20}}{2})$

Step 8.5.3

Add $25$ and $32$ .

$(- \frac{31}{15}, \frac{\frac{57}{20}}{2})$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{31}{15}, \frac{57}{20} \cdot \frac{1}{2})$

Step 10

Multiply $\frac{57}{20} \cdot \frac{1}{2}$ .

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Step 10.1

Multiply $\frac{57}{20}$ by $\frac{1}{2}$ .

$(- \frac{31}{15}, \frac{57}{20 \cdot 2})$

Step 10.2

Multiply $20$ by $2$ .

$(- \frac{31}{15}, \frac{57}{40})$

Step 11