Questions: What is the equation of the line that goes through the points (0,2) and (1, 1)?

Transcript text: What is the equation of the line that goes through the points $(0,2)$ and $(1, 1)$?

Solution

Solution Steps

To find the equation of the line that goes through two points, we need to determine the slope and then use the point-slope form of the line equation. The slope $ m $ is calculated as the change in $ y $ divided by the change in $ x $. Once we have the slope, we can use one of the points to find the y-intercept $ b $ using the equation $ y = mx + b $.

Step 1: Calculate the Slope

To find the equation of the line passing through the points $(0, 2)$ and $(1, 1)$, we first calculate the slope $ m $. The slope is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points: \[ m = \frac{1 - 2}{1 - 0} = \frac{-1}{1} = -1.0 \]

Step 2: Calculate the Y-Intercept

Next, we use the slope $ m $ and one of the points to find the y-intercept $ b $. Using the point $(0, 2)$: \[ y = mx + b \] Substituting $ x = 0 $, $ y = 2 $, and $ m = -1.0 $: \[ 2 = -1.0 \cdot 0 + b \implies b = 2.0 \]

Step 3: Form the Equation of the Line

With the slope $ m = -1.0 $ and the y-intercept $ b = 2.0 $, the equation of the line is: \[ y = -1.0x + 2.0 \]