To find the equation of the second line and the solution of the system of equations, follow these steps:
1. Identify the points provided in the second table:
[tex]\((-6, 9)\)[/tex], [tex]\((-3, 8)\)[/tex], [tex]\( (0, 7) \)[/tex], [tex]\( (3, 6) \)[/tex].
2. Determine the slope [tex]\(m\)[/tex] for the second set of data points. Given that they form a straight line:
- Selecting two points, for example, [tex]\((0, 7)\)[/tex] and [tex]\((3, 6)\)[/tex], we can calculate the slope [tex]\(m\)[/tex] using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For the points [tex]\((0, 7)\)[/tex] and [tex]\((3, 6)\)[/tex]:
[tex]\[
m = \frac{6 - 7}{3 - 0} = \frac{-1}{3} = -\frac{1}{3}
\][/tex]
3. To find the y-intercept [tex]\(b\)[/tex], we use one of the points and the slope formula [tex]\(y = mx + b\)[/tex]. Using the point [tex]\((0, 7)\)[/tex]:
[tex]\[
7 = -\frac{1}{3}(0) + b
\][/tex]
[tex]\[
b = 7
\][/tex]
Therefore, the equation of the second line is:
[tex]\[
y = -\frac{1}{3}x + 7
\][/tex]
4. Given that both equations have the same form, let's verify the unique solution of the system. The equations are:
[tex]\[
\begin{cases}
y = -\frac{1}{3}x + 7 & \text{(Equation 1)} \\
y = -\frac{1}{3}x + 7 & \text{(Equation 2)}
\end{cases}
\][/tex]
Because these are identical equations, any point on this line will satisfy both equations.
5. Therefore, the values obtained are:
- Slope of the second line: [tex]\(-\frac{1}{3}\)[/tex]
- y-intercept of the second line: [tex]\(7\)[/tex]
- Solution to the system: Since both lines overlap entirely, they have infinitely many solutions along the line [tex]\(y = -\frac{1}{3} x + 7\)[/tex].
In summary:
The equation of the second line is [tex]\(y = -\frac{1}{3} x + 7\)[/tex]. This implies the system has infinitely many solutions along that line, not a unique (x, y) pair solution. However, the particular output given suggests a large numerical solution due to computational limitations.
```
Equation of the second line: y = -1/3 x + 7
Solution to the system: (Infinity solutions, line y = -1/3 x + 7)
```
