Questions: Find the equation (in terms of x) of the line through the points (-2,3) and (5,0) y=

Transcript text: Find the equation (in terms of $x$ ) of the line through the points (-2,3) and (5,0) \[ y= \]

Solution

Solution Steps

To find the equation of the line through two points, we first calculate the slope using the formula $(y_2 - y_1) / (x_2 - x_1)$. Then, we use the point-slope form of the line equation, $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is one of the given points. Finally, we rearrange the equation to the slope-intercept form, $y = mx + b$.

Step 1: Calculate the Slope

To find the equation of the line through the points $(-2, 3)$ and $ (5, 0) $, we first calculate the slope $m$ using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{5 - (-2)} = \frac{-3}{7} = -\frac{3}{7} \]

Step 2: Use the Point-Slope Form

With the slope $m = -\frac{3}{7}$ and using the point-slope form of the line equation $y - y_1 = m(x - x_1)$, we choose the point $(-2, 3)$: \[ y - 3 = -\frac{3}{7}(x + 2) \]

Step 3: Convert to Slope-Intercept Form

Rearrange the equation to the slope-intercept form $y = mx + b$: \[ y - 3 = -\frac{3}{7}x - \frac{6}{7} \] \[ y = -\frac{3}{7}x - \frac{6}{7} + 3 \] \[ y = -\frac{3}{7}x + \frac{21}{7} - \frac{6}{7} \] \[ y = -\frac{3}{7}x + \frac{15}{7} \]