Find the Equation Using Two Points (3,5) , (7,2)

$(3, 5)$ , $(7, 2)$

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.

$m = \frac{2 - (5)}{7 - (3)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{2 - 5}{7 - (3)}$

Subtract $5$ from $2$ .

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

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{- 3}{7 - 3}$

Subtract $3$ from $7$ .

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

Move the negative in front of the fraction.

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

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.

$y = (- \frac{3}{4}) \cdot x + b$

Substitute the value of $x$ into the equation.

$y = (- \frac{3}{4}) \cdot (3) + b$

Substitute the value of $y$ into the equation.

$5 = (- \frac{3}{4}) \cdot (3) + b$

Find the value of $b$ .

Rewrite the equation as $- \frac{3}{4} \cdot 3 + b = 5$ .

$- \frac{3}{4} \cdot 3 + b = 5$

Simplify each term.

Multiply $- \frac{3}{4} \cdot 3$ .

Multiply $3$ by $- 1$ .

$- 3 (\frac{3}{4}) + b = 5$

Combine $- 3$ and $\frac{3}{4}$ .

$\frac{- 3 \cdot 3}{4} + b = 5$

Multiply $- 3$ by $3$ .

$\frac{- 9}{4} + b = 5$

Move the negative in front of the fraction.

$- \frac{9}{4} + b = 5$

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

Add $\frac{9}{4}$ to both sides of the equation.

$b = 5 + \frac{9}{4}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$b = 5 \cdot \frac{4}{4} + \frac{9}{4}$

Combine $5$ and $\frac{4}{4}$ .

$b = \frac{5 \cdot 4}{4} + \frac{9}{4}$

Combine the numerators over the common denominator.

$b = \frac{5 \cdot 4 + 9}{4}$

Simplify the numerator.

Multiply $5$ by $4$ .

$b = \frac{20 + 9}{4}$

Add $20$ and $9$ .

$b = \frac{29}{4}$

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.

$y = - \frac{3}{4} x + \frac{29}{4}$

Step 8

Find the Equation Using Two Points (3,5) , (7,2)

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