Find the Equation Using Two Points (2,-2) , (4,1)

$(2, - 2)$ , $(4, 1)$

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.

$m = \frac{1 - (- 2)}{4 - (2)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $- 2$ .

$m = \frac{1 + 2}{4 - (2)}$

Add $1$ and $2$ .

$m = \frac{3}{4 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{3}{4 - 2}$

Subtract $2$ from $4$ .

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

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.

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

Substitute the value of $x$ into the equation.

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

Substitute the value of $y$ into the equation.

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

Find the value of $b$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$3 + b = - 2$

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

Subtract $3$ from both sides of the equation.

$b = - 2 - 3$

Subtract $3$ from $- 2$ .

$b = - 5$

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.

$y = \frac{3}{2} x - 5$

Step 8

Find the Equation Using Two Points (2,-2) , (4,1)

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