Find the Equation Using Two Points (3,4) and (0,5)

$(3, 4)$ and $(0, 5)$

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.

$m = \frac{5 - (4)}{0 - (3)}$

Step 5

Finding the slope $m$ .

Simplify the numerator.

Multiply $- 1$ by $4$ .

$m = \frac{5 - 4}{0 - (3)}$

Subtract $4$ from $5$ .

$m = \frac{1}{0 - (3)}$

Simplify the denominator.

Multiply $- 1$ by $3$ .

$m = \frac{1}{0 - 3}$

Subtract $3$ from $0$ .

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

Move the negative in front of the fraction.

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

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.

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

Substitute the value of $x$ into the equation.

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

Substitute the value of $y$ into the equation.

$4 = (- \frac{1}{3}) \cdot (3) + b$

Find the value of $b$ .

Rewrite the equation as $- \frac{1}{3} \cdot 3 + b = 4$ .

$- \frac{1}{3} \cdot 3 + b = 4$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{- 1}{3} \cdot 3 + b = 4$

Cancel the common factor.

$\frac{- 1}{3} \cdot 3 + b = 4$

Rewrite the expression.

$- 1 + b = 4$

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

Add $1$ to both sides of the equation.

$b = 4 + 1$

Add $4$ and $1$ .

$b = 5$

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.

$y = - \frac{1}{3} x + 5$

Step 8

Find the Equation Using Two Points (3,4) and (0,5)

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