Question: Fill in the missing numbers to complete the linear equation that gives the rule for
this table.
x y
4 2
5 3
6 4
7 5
y = \square x - 2
1. **State the problem:** We are given a table of values for $x$ and $y$ and a linear equation with a missing coefficient: $$y = \square x - 2$$. We need to find the missing number (coefficient) before $x$.
2. **Identify the formula:** The equation is in the form $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. Here, $b = -2$.
3. **Use the table to find the slope $m$:** The slope is the change in $y$ divided by the change in $x$.
Calculate the slope between two points, for example $(4,2)$ and $(5,3)$:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{5 - 4} = \frac{1}{1} = 1$$
4. **Check the slope with another pair to confirm:** Between $(5,3)$ and $(6,4)$:
$$m = \frac{4 - 3}{6 - 5} = \frac{1}{1} = 1$$
5. **Write the equation with the found slope:**
$$y = 1 \times x - 2$$
6. **Verify the equation with a point:** Using $x=4$:
$$y = 1 \times 4 - 2 = 4 - 2 = 2$$ which matches the table.
**Final answer:** The missing number is $1$, so the equation is $$y = 1x - 2$$.