To find the equation of the line that goes through the points in the table (0,8), (3,6), (6,4), (9,2), and (12,0), we first determine the slope (m) using two points. We'll use (0,8) and (3,6) as our points.
The slope \( m \) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 8}{3 - 0} = \frac{-2}{3} \]
Now that we have the slope, we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using the point (3, 6):
\[ y - 6 = -\frac{2}{3}(x - 3) \]
From here, we can rearrange this equation or find the standard form. Let's convert it to slope-intercept form and then to standard form:
Expanding the equation:
\[ y - 6 = -\frac{2}{3}x + 2 \]
Adding 6 to both sides:
\[ y = -\frac{2}{3}x + 8 \]
Now, to convert this to standard form \( Ax + By = C \), we get:
\[ \frac{2}{3}x + y = 8 \]
Multiplying through by 3 to eliminate the fraction:
\[ 2x + 3y = 24 \]
Thus, the standard form of the line is:
2x + 3y = 24.
The correct answer option is:
2x + 3y = 24.
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