The table shows a relationship between $x$ and $y$. Observing the pattern:

• As $x$ increases by 1, $y$ increases by 4.
• This suggests the relationship between $x$ and $y$ is linear.

Deriving the Equation:

1. The general form of a linear equation is: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

2. Calculate the slope ($m$): $m = \frac{\Delta y}{\Delta x} = \frac{11 - 7}{2 - 1} = 4$

3. Substitute $m = 4$ into the equation: $y = 4x + b$

4. Solve for $b$ using a point from the table. For $x = 1, y = 7$: $7 = 4(1) + b \implies b = 3$

5. The equation is: $y = 4x + 3$

Verifying:

For each $x$, substituting into $y = 4x + 3$ gives the correct $y$-values from the table:

• $x = 1$: $y = 4(1) + 3 = 7$
• $x = 2$: $y = 4(2) + 3 = 11$
• $x = 3$: $y = 4(3) + 3 = 15$
• $x = 4$: $y = 4(4) + 3 = 19$

This confirms the equation $y = 4x + 3$.

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Would you like me to explain the derivation in more detail or explore related questions?

Related Questions:

1. What is the value of $y$ when $x = 10$?
2. How can the slope $m$ be interpreted in real-world contexts?
3. What happens if the y-intercept $b$ changes in the equation?
4. Can we plot the line represented by $y = 4x + 3$?
5. What are the domain and range of this linear function?

Tip:

To identify a linear relationship, check if the change in $y$ is consistent for equal changes in $x$.