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