Find the Equation Using Two Points (3,5) , (5,13)

$(3, 5)$ , $(5, 13)$

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.

$m = \frac{13 - (5)}{5 - (3)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{13 - 5}{5 - (3)}$

Subtract $5$ from $13$ .

$m = \frac{8}{5 - (3)}$

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{8}{5 - 3}$

Subtract $3$ from $5$ .

$m = \frac{8}{2}$

Divide $8$ by $2$ .

$m = 4$

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.

$y = (4) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (4) \cdot (3) + b$

Substitute the value of $y$ into the equation.

$5 = (4) \cdot (3) + b$

Find the value of $b$ .

Rewrite the equation as $(4) \cdot (3) + b = 5$ .

$(4) \cdot (3) + b = 5$

Multiply $4$ by $3$ .

$12 + b = 5$

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

Subtract $12$ from both sides of the equation.

$b = 5 - 12$

Subtract $12$ from $5$ .

$b = - 7$

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.

$y = 4 x - 7$

Step 8

Find the Equation Using Two Points (3,5) , (5,13)

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