Find the Equation Using Two Points (-2,3) , (4,5)

$(- 2, 3)$ , $(4, 5)$

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.

$m = \frac{5 - (3)}{4 - (- 2)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{5 - 3}{4 - (- 2)}$

Subtract $3$ from $5$ .

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

Simplify the denominator.

Multiply $- 1$ by $- 2$ .

$m = \frac{2}{4 + 2}$

Add $4$ and $2$ .

$m = \frac{2}{6}$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$m = \frac{2 (1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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.

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

Substitute the value of $x$ into the equation.

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

Substitute the value of $y$ into the equation.

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

Find the value of $b$ .

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

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

Simplify each term.

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

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

Move the negative in front of the fraction.

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

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

Add $\frac{2}{3}$ to both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $3$ .

$b = \frac{9 + 2}{3}$

Add $9$ and $2$ .

$b = \frac{11}{3}$

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.

$y = \frac{1}{3} x + \frac{11}{3}$

Step 8

Find the Equation Using Two Points (-2,3) , (4,5)

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