To find the standard form of the equation of the line that goes through the points given in the table, we need to determine the slope (m) and the y-intercept (b) from the coordinates (x, y). The points from the table are:
- (0, 8)
- (3, 6)
- (6, 4)
- (9, 2)
- (12, 0)
Step 1: Calculate the slope (m)
The slope \( m \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (0, 8) and (3, 6):
\[ m = \frac{6 - 8}{3 - 0} = \frac{-2}{3} \]
Step 2: Write the equation in slope-intercept form (y = mx + b)
Now that we have the slope, we can use one of the points to find the intercept \( b \). We can use the point (0, 8):
\[ 8 = \left(-\frac{2}{3}\right)(0) + b \] \[ b = 8 \]
Thus, the equation of the line in slope-intercept form is:
\[ y = -\frac{2}{3}x + 8 \]
Step 3: Convert to standard form (Ax + By = C)
To convert from slope-intercept form to standard form, we will rearrange the equation:
- Start with: \( y = -\frac{2}{3}x + 8 \)
- Multiply through by 3 to eliminate the fraction: \[ 3y = -2x + 24 \]
- Rearrange into standard form: \[ 2x + 3y = 24 \]
Conclusion
The standard form of the equation of the line going through the points in the table is:
\[ \boxed{2x + 3y = 24} \]
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