To determine the standard form of the equation of the line passing through the points in the table, we'll first calculate the slope from the given points and then use the point-slope form to derive the equation.
Points from the Table
The points provided are:
- (3, 11)
- (7, 8)
- (11, 5)
- (15, 2)
- (19, -1)
Step 1: Calculate the slope (m)
We can use any two points to find the slope. Let's use the points (3, 11) and (7, 8).
The formula for the slope (m) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using points (3, 11) and (7, 8): \[ m = \frac{8 - 11}{7 - 3} = \frac{-3}{4} \]
Step 2: Point-Slope Form
Now, we can use the point-slope form of the equation: \[ y - y_1 = m(x - x_1) \] Using point (3, 11): \[ y - 11 = -\frac{3}{4}(x - 3) \]
Step 3: Convert to Standard Form
Now we'll convert \(y - 11 = -\frac{3}{4}(x - 3)\) into standard form. This gives: \[ y - 11 = -\frac{3}{4}x + \frac{9}{4} \] Multiply everything by 4 to eliminate the fraction: \[ 4y - 44 = -3x + 9 \] Rearranging gives: \[ 3x + 4y = 53 \]
Conclusion
The standard form of the equation of the line going through the points in the table is: \[ 3x + 4y = 53 \]
Thus, the correct response is: 3x + 4y = 53
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