Examples

(\frac{1}{3}, \frac{4}{7})

$(\frac{1}{3}, \frac{4}{7})$ ,

(1, 0)

$(1, 0)$

Step 1

Use

y = m x + b

$y = m x + b$ to calculate the equation of the line, where

m

$m$ represents the slope and

b

$b$ represents the y-intercept.

To calculate the equation of the line, use the

y = m x + b

$y = m x + b$ format.

Step 2

Slope is equal to the change in

y

$y$ over the change in

x

$x$ , or rise over run.

m = \frac{(change in y)}{(change in x)}

$m = \frac{(change in y)}{(change in x)}$

Step 3

The change in

x

$x$ is equal to the difference in x-coordinates (also called run), and the change in

y

$y$ is equal to the difference in y-coordinates (also called rise).

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 4

Substitute in the values of

x

$x$ and

y

$y$ into the equation to find the slope.

m = \frac{0 - (\frac{4}{7})}{1 - (\frac{1}{3})}

$m = \frac{0 - (\frac{4}{7})}{1 - (\frac{1}{3})}$

Step 5

Finding the slope

m

$m$ .

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Step 5.1

Multiply the numerator and denominator of the fraction by

21

$21$ .

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Step 5.1.1

Multiply

\frac{0 - \frac{4}{7}}{1 - \frac{1}{3}}

$\frac{0 - \frac{4}{7}}{1 - \frac{1}{3}}$ by

\frac{21}{21}

$\frac{21}{21}$ .

m = \frac{21}{21} \cdot \frac{0 - \frac{4}{7}}{1 - \frac{1}{3}}

$m = \frac{21}{21} \cdot \frac{0 - \frac{4}{7}}{1 - \frac{1}{3}}$

Step 5.1.2

Combine.

m = \frac{21 (0 - \frac{4}{7})}{21 (1 - \frac{1}{3})}

$m = \frac{21 (0 - \frac{4}{7})}{21 (1 - \frac{1}{3})}$

m = \frac{21 (0 - \frac{4}{7})}{21 (1 - \frac{1}{3})}

$m = \frac{21 (0 - \frac{4}{7})}{21 (1 - \frac{1}{3})}$

Step 5.2

Apply the distributive property.

m = \frac{21 \cdot 0 + 21 (- \frac{4}{7})}{21 \cdot 1 + 21 (- \frac{1}{3})}

$m = \frac{21 \cdot 0 + 21 (- \frac{4}{7})}{21 \cdot 1 + 21 (- \frac{1}{3})}$

Step 5.3

Simplify by cancelling.

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Step 5.3.1

Cancel the common factor of

7

$7$ .

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Step 5.3.1.1

Move the leading negative in

- \frac{4}{7}

$- \frac{4}{7}$ into the numerator.

m = \frac{21 \cdot 0 + 21 (\frac{- 4}{7})}{21 \cdot 1 + 21 (- \frac{1}{3})}

$m = \frac{21 \cdot 0 + 21 (\frac{- 4}{7})}{21 \cdot 1 + 21 (- \frac{1}{3})}$

Step 5.3.1.2

Factor

7

$7$ out of

21

$21$ .

m = \frac{21 \cdot 0 + 7 (3) (\frac{- 4}{7})}{21 \cdot 1 + 21 (- \frac{1}{3})}

$m = \frac{21 \cdot 0 + 7 (3) (\frac{- 4}{7})}{21 \cdot 1 + 21 (- \frac{1}{3})}$

Step 5.3.1.3

Cancel the common factor.

$m = \frac{21 \cdot 0 + 7 \cdot (3 (\frac{- 4}{7}))}{21 \cdot 1 + 21 (- \frac{1}{3})}$

Step 5.3.1.4

Rewrite the expression.

$m = \frac{21 \cdot 0 + 3 \cdot - 4}{21 \cdot 1 + 21 (- \frac{1}{3})}$

Step 5.3.2

Multiply

$3$ by

$- 4$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + 21 (- \frac{1}{3})}$

Step 5.3.3

Cancel the common factor of

$3$ .

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Step 5.3.3.1

Move the leading negative in

$- \frac{1}{3}$ into the numerator.

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + 21 (\frac{- 1}{3})}$

Step 5.3.3.2

Factor

$3$ out of

$21$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + 3 (7) (\frac{- 1}{3})}$

Step 5.3.3.3

Cancel the common factor.

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + 3 \cdot (7 (\frac{- 1}{3}))}$

Step 5.3.3.4

Rewrite the expression.

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 + 7 \cdot - 1}$

Step 5.3.4

Multiply

$7$ by

$- 1$ .

$m = \frac{21 \cdot 0 - 12}{21 \cdot 1 - 7}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply

$21$ by

$0$ .

$m = \frac{0 - 12}{21 \cdot 1 - 7}$

Step 5.4.2

Subtract

$12$ from

$0$ .

$m = \frac{- 12}{21 \cdot 1 - 7}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply

$21$ by

$1$ .

$m = \frac{- 12}{21 - 7}$

Step 5.5.2

Subtract

$7$ from

$21$ .

$m = \frac{- 12}{14}$

Step 5.6

Reduce the expression by cancelling the common factors.

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Step 5.6.1

Cancel the common factor of

$- 12$ and

$14$ .

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Step 5.6.1.1

Factor

$2$ out of

$- 12$ .

$m = \frac{2 (- 6)}{14}$

Step 5.6.1.2

Cancel the common factors.

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Step 5.6.1.2.1

Factor

$2$ out of

$14$ .

$m = \frac{2 \cdot - 6}{2 \cdot 7}$

Step 5.6.1.2.2

Cancel the common factor.

$m = \frac{2 \cdot - 6}{2 \cdot 7}$

Step 5.6.1.2.3

Rewrite the expression.

$m = \frac{- 6}{7}$

Step 5.6.2

Move the negative in front of the fraction.

$m = - \frac{6}{7}$

Step 6

Find the value of

$b$ using the formula for the equation of a line.

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Step 6.1

Use the formula for the equation of a line to find

$b$ .

$y = m x + b$

Step 6.2

Substitute the value of

$m$ into the equation.

$y = (- \frac{6}{7}) \cdot x + b$

Step 6.3

Substitute the value of

$x$ into the equation.

$y = (- \frac{6}{7}) \cdot (\frac{1}{3}) + b$

Step 6.4

Substitute the value of

$y$ into the equation.

$\frac{4}{7} = (- \frac{6}{7}) \cdot (\frac{1}{3}) + b$

Step 6.5

Find the value of

$b$ .

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Step 6.5.1

Rewrite the equation as

$- \frac{6}{7} \cdot \frac{1}{3} + b = \frac{4}{7}$ .

$- \frac{6}{7} \cdot \frac{1}{3} + b = \frac{4}{7}$

Step 6.5.2

Simplify each term.

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Step 6.5.2.1

Cancel the common factor of

$3$ .

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Step 6.5.2.1.1

Move the leading negative in

$- \frac{6}{7}$ into the numerator.

$\frac{- 6}{7} \cdot \frac{1}{3} + b = \frac{4}{7}$

Step 6.5.2.1.2

Factor

$3$ out of

$- 6$ .

$\frac{3 (- 2)}{7} \cdot \frac{1}{3} + b = \frac{4}{7}$

Step 6.5.2.1.3

Cancel the common factor.

$\frac{3 \cdot - 2}{7} \cdot \frac{1}{3} + b = \frac{4}{7}$

Step 6.5.2.1.4

Rewrite the expression.

$\frac{- 2}{7} + b = \frac{4}{7}$

Step 6.5.2.2

Move the negative in front of the fraction.

$- \frac{2}{7} + b = \frac{4}{7}$

Step 6.5.3

Move all terms not containing

$b$ to the right side of the equation.

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Step 6.5.3.1

Add

$\frac{2}{7}$ to both sides of the equation.

$b = \frac{4}{7} + \frac{2}{7}$

Step 6.5.3.2

Combine the numerators over the common denominator.

$b = \frac{4 + 2}{7}$

Step 6.5.3.3

Add

$4$ and

$2$ .

$b = \frac{6}{7}$

Step 7

Now that the values of

$m$ (slope) and

$b$ (y-intercept) are known, substitute them into

$y = m x + b$ to find the equation of the line.

$y = - \frac{6}{7} x + \frac{6}{7}$

Step 8

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