To find the linear equation that fits the given points, we can start by determining the slope (\(m\)) and y-intercept (\(b\)) of the line in the slope-intercept form, which is \(y = mx + b\).
The points given are:
- (3, -5)
- (4, -10)
- (5, 6)
- (-15, -20)
Step 1: Calculate the slope \(m\)
We can use two points to calculate the slope. Let's use the points (3, -5) and (5, 6):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-5)}{5 - 3} = \frac{6 + 5}{5 - 3} = \frac{11}{2} \]
Thus, the slope \(m\) is \(\frac{11}{2}\).
Step 2: Use the slope and a point to find the y-intercept \(b\)
We can use the point (3, -5) to find \(b\): \[ y = mx + b \] Substituting \(x = 3\) and \(y = -5\): \[ -5 = \frac{11}{2}(3) + b \] Calculating: \[ -5 = \frac{33}{2} + b \] Subtract \(\frac{33}{2}\) from both sides: \[ b = -5 - \frac{33}{2} = -\frac{10}{2} - \frac{33}{2} = -\frac{43}{2} \]
Step 3: Write the final linear equation
Now we have the slope \(m = \frac{11}{2}\) and the y-intercept \(b = -\frac{43}{2}\). Hence, the linear equation is:
\[ y = \frac{11}{2}x - \frac{43}{2} \]
So the linear equation that gives the rule for the table is:
\[ y = \frac{11}{2}x - \frac{43}{2} \]