Find the Slope and y-intercept (2,1) , (5,3)

$(2, 1)$ , $(5, 3)$

Find the value of the slope.

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

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

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

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

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{3 - (1)}{5 - (2)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $1$ .

$m = \frac{3 - 1}{5 - (2)}$

Subtract $1$ from $3$ .

$m = \frac{2}{5 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{2}{5 - 2}$

Subtract $2$ from $5$ .

$m = \frac{2}{3}$

Find the value of the y-intercept.

Substitute the value of $m$ into the slope-intercept form of the equation, $y = m x + b$ .

$y = (\frac{2}{3}) x + b$

Substitute the value of $x$ into the slope-intercept form of the equation, $y = m x + b$ .

$y = (\frac{2}{3}) \cdot (2) + b$

Substitute the value of $y$ into the slope-intercept form of the equation, $y = m x + b$ .

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

Rewrite the equation as $(\frac{2}{3}) \cdot (2) + b = 1$ .

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

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

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

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

Multiply $2$ by $2$ .

$\frac{4}{3} + b = 1$

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

Subtract $\frac{4}{3}$ from both sides of the equation.

$b = 1 - \frac{4}{3}$

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

$b = \frac{3}{3} - \frac{4}{3}$

Combine the numerators over the common denominator.

$b = \frac{3 - 4}{3}$

Subtract $4$ from $3$ .

$b = \frac{- 1}{3}$

Move the negative in front of the fraction.

$b = - \frac{1}{3}$

List the slope and y-intercept.

Slope: $\frac{2}{3}$

y-intercept: $(0, - \frac{1}{3})$

Find the Slope and y-intercept (2,1) , (5,3)

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