Find the Equation Using Two Points (3,1) , (9,4)

$(3, 1)$ , $(9, 4)$

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.

$m = \frac{4 - (1)}{9 - (3)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $1$ .

$m = \frac{4 - 1}{9 - (3)}$

Subtract $1$ from $4$ .

$m = \frac{3}{9 - (3)}$

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{3}{9 - 3}$

Subtract $3$ from $9$ .

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

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

$m = \frac{3 \cdot 1}{3 \cdot 2}$

Cancel the common factor.

$m = \frac{3 \cdot 1}{3 \cdot 2}$

Rewrite the expression.

$m = \frac{1}{2}$

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.

$y = (\frac{1}{2}) x + b$

Substitute the value of $x$ into the equation.

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

Substitute the value of $y$ into the equation.

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

Find the value of $b$ .

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

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

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

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

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

Subtract $\frac{3}{2}$ from both sides of the equation.

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

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

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

Combine the numerators over the common denominator.

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

Subtract $3$ from $2$ .

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

Move the negative in front of the fraction.

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

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.

$y = \frac{1}{2} x - \frac{1}{2}$

Step 8

Find the Equation Using Two Points (3,1) , (9,4)

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