The problem provides a table of $x$ and $y$ values that satisfy a linear equation of the form $y = mx + c$, where $m$ and $c$ are constants. The goal is to find the equation of the line described by the table.

1. Problem Description

The problem provides a table of $x$ and $y$ values that satisfy a linear equation of the form $y = mx + c$, where $m$ and $c$ are constants. The goal is to find the equation of the line described by the table.

2. Solution Steps

The given equation is $y = mx + c$.

We can use two points from the table to determine the values of $m$ and $c$. Let's use the points $(0, 1)$ and $(2, 2)$.

Using the point $(0, 1)$:

$1 = m(0) + c$

$1 = 0 + c$

$c = 1$

Now that we know $c = 1$, the equation is $y = mx + 1$. We can use the point $(2, 2)$ to find $m$:

$2 = m(2) + 1$

$2 - 1 = 2m$

$1 = 2m$

$m = \frac{1}{2}$

So the equation of the line is $y = \frac{1}{2}x + 1$. Let's check if this equation holds true for the third point $(4, 3)$:

$y = \frac{1}{2}(4) + 1$

$y = 2 + 1$

$y = 3$

Since the equation holds true for all three points, the equation of the line is $y = \frac{1}{2}x + 1$.

3. Final Answer

D. $y = \frac{1}{2}x + 1$