Find the Equation Using Two Points (4,5) , (12,9)

$(4, 5)$ , $(12, 9)$

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.

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{(change in y)}{(change in x)}$

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).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{9 - (5)}{12 - (4)}$

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $5$ .

$m = \frac{9 - 5}{12 - (4)}$

Subtract $5$ from $9$ .

$m = \frac{4}{12 - (4)}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$m = \frac{4}{12 - 4}$

Subtract $4$ from $12$ .

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

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$m = \frac{4 (1)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$m = \frac{4 \cdot 1}{4 \cdot 2}$

Cancel the common factor.

$m = \frac{4 \cdot 1}{4 \cdot 2}$

Rewrite the expression.

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

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.

$y = (\frac{1}{2}) x + b$

Substitute the value of $x$ into the equation.

$y = (\frac{1}{2}) \cdot (4) + b$

Substitute the value of $y$ into the equation.

$5 = (\frac{1}{2}) \cdot (4) + b$

Find the value of $b$ .

Rewrite the equation as $\frac{1}{2} \cdot 4 + b = 5$ .

$\frac{1}{2} \cdot 4 + b = 5$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{1}{2} \cdot (2 (2)) + b = 5$

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 2) + b = 5$

Rewrite the expression.

$2 + b = 5$

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

Subtract $2$ from both sides of the equation.

$b = 5 - 2$

Subtract $2$ from $5$ .

$b = 3$

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.

$y = \frac{1}{2} x + 3$

Find the Equation Using Two Points (4,5) , (12,9)

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