Find the Equation Using Two Points (1,4) , (3,5)

$(1, 4)$ , $(3, 5)$

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.

$m = \frac{5 - (4)}{3 - (1)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $4$ .

$m = \frac{5 - 4}{3 - (1)}$

Subtract $4$ from $5$ .

$m = \frac{1}{3 - (1)}$

Simplify the denominator.

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $3$ .

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

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.

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

Substitute the value of $x$ into the equation.

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

Substitute the value of $y$ into the equation.

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

Find the value of $b$ .

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

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

Multiply $\frac{1}{2}$ by $1$ .

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

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

Subtract $\frac{1}{2}$ from both sides of the equation.

$b = 4 - \frac{1}{2}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = 4 \cdot \frac{2}{2} - \frac{1}{2}$

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

$b = \frac{4 \cdot 2}{2} - \frac{1}{2}$

Combine the numerators over the common denominator.

$b = \frac{4 \cdot 2 - 1}{2}$

Simplify the numerator.

Multiply $4$ by $2$ .

$b = \frac{8 - 1}{2}$

Subtract $1$ from $8$ .

$b = \frac{7}{2}$

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.

$y = \frac{1}{2} x + \frac{7}{2}$

Step 8

Find the Equation Using Two Points (1,4) , (3,5)

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