To find the equation of the line that goes through the points given in the table, we can first determine the slope (m) and then use the point-slope form of a line equation to convert it to standard form.
The points given in the table are:
- (3, 11)
- (7, 8)
- (11, 5)
- (15, 2)
- (19, -1)
Step 1: Calculate the slope (m) Using two points, for example, (3, 11) and (7, 8):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 11}{7 - 3} = \frac{-3}{4} \]
So the slope \( m = -\frac{3}{4} \).
Step 2: Use the point-slope form We can use the point-slope form of the line equation, which is given by: \[ y - y_1 = m(x - x_1) \] Using the point (3, 11): \[ y - 11 = -\frac{3}{4}(x - 3) \]
Step 3: Distribute and rearrange to get into slope-intercept form \[ y - 11 = -\frac{3}{4}x + \frac{9}{4} \] \[ y = -\frac{3}{4}x + \frac{9}{4} + 11 \] Convert \( 11 \) to fourths to simplify: \[ y = -\frac{3}{4}x + \frac{9}{4} + \frac{44}{4} = -\frac{3}{4}x + \frac{53}{4} \]
Step 4: Convert to standard form \(Ax + By = C\) Multiply the entire equation by 4 to eliminate the fraction: \[ 4y = -3x + 53 \] Rearranging gives: \[ 3x + 4y = 53 \]
Thus, the standard form of the equation of the line is: \[ 3x + 4y = 53 \]
So the answer is \( 3x + 4y = 53 \).
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