Write the equation for the line perpendicular to the line in the table that goes through point (6,3).

Give your answer in Slope-Intercept Form.

(table values:
x - 2, 4, 6, 8
y - 5, 2, -1, -4)

1 answer

First, we need to determine the slope of the line represented by the points in the table. The points given are (2, 5), (4, 2), (6, -1), and (8, -4).

We can calculate the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's take the first two points (2, 5) and (4, 2):

\[ m = \frac{2 - 5}{4 - 2} = \frac{-3}{2} \]

Now, we can find the slope of the line perpendicular to this line. The slope of a line perpendicular to another is the negative reciprocal of the original slope.

So, the slope of the perpendicular line is:

\[ m_{perpendicular} = -\frac{1}{m} = -\frac{1}{-\frac{3}{2}} = \frac{2}{3} \]

Now, we need to use this slope and the given point (6, 3) to write the equation of the line in slope-intercept form (y = mx + b).

Using the point-slope formula:

\[ y - y_1 = m(x - x_1) \]

We can substitute \( m \), \( x_1 \), and \( y_1 \):

\[ y - 3 = \frac{2}{3}(x - 6) \]

Now, we can simplify this equation:

\[ y - 3 = \frac{2}{3}x - \frac{2}{3} \cdot 6 \] \[ y - 3 = \frac{2}{3}x - 4 \] \[ y = \frac{2}{3}x - 4 + 3 \] \[ y = \frac{2}{3}x - 1 \]

Thus, the equation of the line perpendicular to the given line and passing through the point (6, 3) in slope-intercept form is:

\[ \boxed{y = \frac{2}{3}x - 1} \]