Identify the Equation that Matches the Function in the Table of X and Y

To determine the correct equation for the function represented by the table, we will follow these steps:

• Step 1: Identify the pattern by calculating the slope $m$.

• Step 2: Determine the y-intercept $c$.

• Step 3: Select the equation that consistently matches all pairs in the table.

Step 1: Finding the slope $m$
The slope $m$ of a linear function $y = mx + c$ is given by the change in $Y$ over the change in $X$. Let's compute the slope between any two points from the table, say between $(0, -5)$ and $(1, -3)$:

$m = \frac{\Delta Y}{\Delta X} = \frac{-3 - (-5)}{1 - 0} = \frac{2}{1} = 2$

Step 2: Determining the y-intercept $c$
Using the point $(0, -5)$, we can find the y-intercept directly since $X = 0$:

$Y = 2 \times 0 + c = -5$

So, $c = -5$.

Step 3: Verifying Equation Match
The linear equation becomes $y = 2x - 5$. Substitute the remaining pairs:

- For $X = 1$, $Y = 2(1) - 5 = 2 - 5 = -3$

- For $X = 2$, $Y = 2(2) - 5 = 4 - 5 = -1$

- For $X = 3$, $Y = 2(3) - 5 = 6 - 5 = 1$

- For $X = 4$, $Y = 2(4) - 5 = 8 - 5 = 3$

All pairs check out, confirming $y = 2x - 5$ aligns perfectly.

Therefore, the solution to the problem is $y = 2x - 5$.