Find the Equation Using Two Points (3,-1) , (4,7)

$(3, - 1)$ , $(4, 7)$

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.

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

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $- 1$ .

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

Add $7$ and $1$ .

$m = \frac{8}{4 - (3)}$

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{8}{4 - 3}$

Subtract $3$ from $4$ .

$m = \frac{8}{1}$

Divide $8$ by $1$ .

$m = 8$

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.

$y = (8) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (8) \cdot (3) + b$

Substitute the value of $y$ into the equation.

$- 1 = (8) \cdot (3) + b$

Find the value of $b$ .

Rewrite the equation as $(8) \cdot (3) + b = - 1$ .

$(8) \cdot (3) + b = - 1$

Multiply $8$ by $3$ .

$24 + b = - 1$

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

Subtract $24$ from both sides of the equation.

$b = - 1 - 24$

Subtract $24$ from $- 1$ .

$b = - 25$

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.

$y = 8 x - 25$

Step 8

Find the Equation Using Two Points (3,-1) , (4,7)

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