Questions: Find an equation for the line that passes through the points (-5,3) and (4,6).

Solution Steps

To find the equation of the line that passes through two points, we can use the point-slope form of the equation of a line. The steps are as follows:

1. Calculate the slope (m) using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
2. Use the point-slope form $y - y_1 = m(x - x_1)$ with one of the given points to find the equation of the line.

To find the slope $m$ of the line that passes through the points $(-5, 3)$ and $(4, 6)$, we use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 3}{4 - (-5)} = \frac{3}{9} = \frac{1}{3}$$

Next, we apply the point-slope form of the equation of a line, which is given by:

$$y - y_1 = m(x - x_1)$$

Using the point $(-5, 3)$ and the slope $m = \frac{1}{3}$:

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

Now, we rearrange the equation to the slope-intercept form $y = mx + b$:

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

Distributing the slope:

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

Adding 3 to both sides:

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

Converting 3 to a fraction with a common denominator:

$$3 = \frac{9}{3}$$

Thus, we have:

$$y = \frac{1}{3}x + \frac{5}{3} + \frac{9}{3} = \frac{1}{3}x + \frac{14}{3}$$

Final Answer