To determine the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] given the data points, we need to find the equation of the line that best fits these points. The equation of a line is generally given by [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
Here are the given points:
[tex]\[
\begin{tabular}{l|rl}
$x$ & $y$ \\
\hline
0 & -6 \\
1 & 1 \\
2 & 8 \\
3 & 15 \\
4 & 22 \\
5 & 29 \\
\end{tabular}
\][/tex]
### Step 1: Finding the Slope ([tex]\(m\)[/tex])
The slope of the line, [tex]\(m\)[/tex], can be interpreted as the change in [tex]\(y\)[/tex] for a unit change in [tex]\(x\)[/tex]. From the points, we can observe the changes in [tex]\(y\)[/tex]:
- From [tex]\(x = 0\)[/tex] to [tex]\(x = 1\)[/tex]: [tex]\(y\)[/tex] changes from [tex]\(-6\)[/tex] to [tex]\(1\)[/tex], a change of [tex]\(7\)[/tex].
- From [tex]\(x = 1\)[/tex] to [tex]\(x = 2\)[/tex]: [tex]\(y\)[/tex] changes from [tex]\(1\)[/tex] to [tex]\(8\)[/tex], a change of [tex]\(7\)[/tex].
- Similarly, the change from [tex]\(x = 2\)[/tex] to [tex]\(x = 3\)[/tex], from [tex]\(x = 3\)[/tex] to [tex]\(x = 4\)[/tex], and from [tex]\(x = 4\)[/tex] to [tex]\(x = 5\)[/tex] also shows a change of [tex]\(7\)[/tex].
Thus, the changes in [tex]\(y\)[/tex] are consistent and indicate a slope, [tex]\(m\)[/tex], of [tex]\(7\)[/tex].
### Step 2: Finding the Y-Intercept ([tex]\(b\)[/tex])
To find the y-intercept, [tex]\(b\)[/tex], we use the point where [tex]\(x = 0\)[/tex] and [tex]\(y = -6\)[/tex].
Since [tex]\(y = mx + b\)[/tex]:
[tex]\[ y = 7x + b \][/tex]
When [tex]\(x = 0\)[/tex]:
[tex]\[ -6 = 7 \cdot 0 + b \][/tex]
[tex]\[ b = -6 \][/tex]
### Step 3: Writing the Equation
Now we can write the equation of the line that fits the given points. Substituting [tex]\(m = 7\)[/tex] and [tex]\(b = -6\)[/tex] into the equation [tex]\(y = mx + b\)[/tex], we get:
[tex]\[ y = 7x - 6 \][/tex]
### Step 4: Verifying with the Given Data Points
To ensure the equation [tex]\(y = 7x - 6\)[/tex] fits all the given points, we can check by plugging in the values of [tex]\(x\)[/tex]:
- For [tex]\(x = 0\)[/tex]:
[tex]\[ y = 7(0) - 6 = -6 \][/tex]
- For [tex]\(x = 1\)[/tex]:
[tex]\[ y = 7(1) - 6 = 1 \][/tex]
- For [tex]\(x = 2\)[/tex]:
[tex]\[ y = 7(2) - 6 = 14 - 6 = 8 \][/tex]
- For [tex]\(x = 3\)[/tex]:
[tex]\[ y = 7(3) - 6 = 21 - 6 = 15 \][/tex]
- For [tex]\(x = 4\)[/tex]:
[tex]\[ y = 7(4) - 6 = 28 - 6 = 22 \][/tex]
- For [tex]\(x = 5\)[/tex]:
[tex]\[ y = 7(5) - 6 = 35 - 6 = 29 \][/tex]
All points fit perfectly with the derived equation.
### Conclusion
The relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is described by the equation:
[tex]\[ y = 7x - 6 \][/tex]
