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

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

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.

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

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $7$ .

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

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{4}{5 - 2}$

Subtract $2$ from $5$ .

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

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.

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

Substitute the value of $x$ into the equation.

$y = (\frac{4}{3}) \cdot (2) + b$

Substitute the value of $y$ into the equation.

$3 = (\frac{4}{3}) \cdot (2) + b$

Find the value of $b$ .

Rewrite the equation as $\frac{4}{3} \cdot 2 + b = 3$ .

$\frac{4}{3} \cdot 2 + b = 3$

Multiply $\frac{4}{3} \cdot 2$ .

Combine $\frac{4}{3}$ and $2$ .

$\frac{4 \cdot 2}{3} + b = 3$

Multiply $4$ by $2$ .

$\frac{8}{3} + b = 3$

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

Subtract $\frac{8}{3}$ from both sides of the equation.

$b = 3 - \frac{8}{3}$

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

$b = 3 \cdot \frac{3}{3} - \frac{8}{3}$

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

$b = \frac{3 \cdot 3}{3} - \frac{8}{3}$

Combine the numerators over the common denominator.

$b = \frac{3 \cdot 3 - 8}{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$b = \frac{9 - 8}{3}$

Subtract $8$ from $9$ .

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

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.

$y = \frac{4}{3} x + \frac{1}{3}$

Step 8

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

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