The table in the image shows two columns, $x$ and $y$, where each $y$ corresponds to a certain $x$. Let us analyze the pattern to determine the rule or formula for $y$ in terms of $x$.

Table:

Step 1: Look for a relationship

Observing the values in $y$ for consecutive $x$ values:

• From $x = 0$ to $x = 1$: $y$ increases from 1 to 5 ($+4$).
• From $x = 1$ to $x = 2$: $y$ increases from 5 to 9 ($+4$).
• From $x = 2$ to $x = 3$: $y$ increases from 9 to 13 ($+4$).

The increase is constant ($+4$), so $y$ follows a linear relationship of the form: $y = mx + c$ where $m$ is the slope, and $c$ is the y-intercept.

Step 2: Calculate the slope ($m$)

Using two points, such as $(0, 1)$ and $(1, 5)$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 1}{1 - 0} = 4$

Thus, $m = 4$.

Step 3: Find the y-intercept ($c$)

Using the point $(0, 1)$ (where $x = 0$, $y = 1$): $y = mx + c \implies 1 = 4(0) + c \implies c = 1$

Step 4: Write the rule

The rule for $y$ is: $y = 4x + 1$

Step 5: Complete the missing values

• When $x = 10$: $y = 4(10) + 1 = 40 + 1 = 41$
• To find $x$ when $y = ?$, you would substitute $y$ into the formula and solve for $x$.

Would you like more clarification on this or how to handle other values? Let me know!

Related Questions:

1. How can you find the equation of a line given two points?
2. What is the significance of the slope and y-intercept in a linear equation?
3. How can you extend this rule for non-integer values of $x$?
4. What happens if the difference between $y$ values is not constant?
5. How can you graph this linear equation?

Tip: Always look for patterns in differences when identifying relationships in tables!