To find the equation represented by the second table, we will first identify the slope and the y-intercept from the given points.
The points in the second table are:
- (-6, 9)
- (-3, 8)
- (0, 7)
- (3, 6)
We can use any two points to find the slope \( m \). Let's use the points (0, 7) and (3, 6):
- Calculate the slope \( m \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 7}{3 - 0} = \frac{-1}{3} \]
- Now that we have the slope, we can use the point-slope form of the line equation to find the y-intercept. We can use the point (0, 7):
The equation in slope-intercept form is: \[ y = mx + b \] Substituting for \( m \) and the point: \[ 7 = \left(-\frac{1}{3}\right)(0) + b \implies b = 7 \]
Now we can write the entire equation for the second line: \[ y = -\frac{1}{3}x + 7 \]
Now, we can find the solution of the system of equations by solving the equations together. The first equation is not given explicitly, but we can infer it based on the points provided in the first table, which suggests a line that appears to have a slope of 1 and a y-intercept of 5.
The first line's equation looks like this: \[ y = x + 5 \]
To find the solution, set both equations equal: \[ x + 5 = -\frac{1}{3}x + 7 \]
Solve for \( x \): \[ x + \frac{1}{3}x = 7 - 5 \] \[ \frac{4}{3}x = 2 \] \[ x = 2 \cdot \frac{3}{4} = \frac{3}{2} \]
Now substitute \( x = \frac{3}{2} \) back to find \( y \): \[ y = x + 5 = \frac{3}{2} + 5 = \frac{3}{2} + \frac{10}{2} = \frac{13}{2} \]
Thus, the solution of the system is: \[ \left(\frac{3}{2}, \frac{13}{2}\right) \]
So the final answer is: The equation that represents the other equation is \( y = -\frac{1}{3}x + 7 \). The solution of the system is \(\left(\frac{3}{2}, \frac{13}{2}\right)\).
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