Question: Which equation gives the rule for this table?
x
y
1
8
2
9
3
10
4
11
y = x + 7
y = 6x + 7
y = 2x + 7
y = 23x + 7
1. **State the problem:** We are given a table of values for $x$ and $y$ and need to find the equation that relates $y$ to $x$.
2. **Given table:**
$$\begin{array}{c|c}
x & y \\\hline
1 & 8 \\
2 & 9 \\
3 & 10 \\
4 & 11
\end{array}$$
3. **Identify the type of relationship:** The $y$ values increase by 1 as $x$ increases by 1, suggesting a linear relationship of the form:
$$y = mx + b$$
where $m$ is the slope and $b$ is the y-intercept.
4. **Calculate the slope $m$:**
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 8}{2 - 1} = \frac{1}{1} = 1$$
5. **Find the y-intercept $b$:** Use one point, for example $(1,8)$:
$$8 = 1 \times 1 + b \implies b = 8 - 1 = 7$$
6. **Write the equation:**
$$y = 1x + 7$$
or simply
$$y = x + 7$$
7. **Verify with other points:** For $x=3$,
$$y = 3 + 7 = 10$$
which matches the table.
**Final answer:**
$$y = x + 7$$