Find the Equation Using Two Points (-3,2) , (6,3)

$(- 3, 2)$ , $(6, 3)$

Step 1

Use $y = m x + b$ to calculate the equation of the line, where $m$ represents the slope and $b$ represents the y-intercept.

To calculate the equation of the line, use the $y = m x + b$ format.

Step 2

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

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

Step 3

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).

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

Step 4

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

$m = \frac{3 - (2)}{6 - (- 3)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $2$ .

$m = \frac{3 - 2}{6 - (- 3)}$

Subtract $2$ from $3$ .

$m = \frac{1}{6 - (- 3)}$

Simplify the denominator.

Multiply $- 1$ by $- 3$ .

$m = \frac{1}{6 + 3}$

Add $6$ and $3$ .

$m = \frac{1}{9}$

Step 6

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

Use the formula for the equation of a line to find $b$ .

$y = m x + b$

Substitute the value of $m$ into the equation.

$y = (\frac{1}{9}) x + b$

Substitute the value of $x$ into the equation.

$y = (\frac{1}{9}) \cdot (- 3) + b$

Substitute the value of $y$ into the equation.

$2 = (\frac{1}{9}) \cdot (- 3) + b$

Find the value of $b$ .

Rewrite the equation as $\frac{1}{9} \cdot - 3 + b = 2$ .

$\frac{1}{9} \cdot - 3 + b = 2$

Simplify each term.

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{1}{3}$ and $- 1$ .

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

Move the negative in front of the fraction.

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

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

Add $\frac{1}{3}$ to both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$b = \frac{6 + 1}{3}$

Add $6$ and $1$ .

$b = \frac{7}{3}$

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.

$y = \frac{1}{9} x + \frac{7}{3}$

Step 8

Find the Equation Using Two Points (-3,2) , (6,3)

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