The problem in the image asks if the function $y = -\frac{2}{3}x + \frac{1}{9}$ is a solution to the differential equation $y' = 2x + 3y - 1$.

To determine if this is a solution, we will:

1. Differentiate $y = -\frac{2}{3}x + \frac{1}{9}$ to find $y'$.
2. Substitute $y$ and $y'$ into the given equation $y' = 2x + 3y - 1$ and check if both sides are equal.

Let's start by differentiating $y = -\frac{2}{3}x + \frac{1}{9}$:

$\frac{dy}{dx} = -\frac{2}{3}$

Now, substitute $y = -\frac{2}{3}x + \frac{1}{9}$ and $y' = -\frac{2}{3}$ into the differential equation:

$y' = 2x + 3y - 1$

Substituting:

$-\frac{2}{3} = 2x + 3\left( -\frac{2}{3}x + \frac{1}{9} \right) - 1$

Simplifying the right side:

$-\frac{2}{3} = 2x - 2x + \frac{1}{3} - 1$

$-\frac{2}{3} = \frac{1}{3} - 1$

$-\frac{2}{3} = -\frac{2}{3}$

Since both sides are equal, the function $y = -\frac{2}{3}x + \frac{1}{9}$ is indeed a solution to the given differential equation.

Thus, the correct answer is: A) Yes

Do you want further explanation or details on the steps? Here are some additional questions that may be helpful:

1. What is the general method for solving first-order linear differential equations?
2. How would the solution change if the equation was non-linear?
3. How do we determine if a given function is a solution to a differential equation?
4. What is the significance of differentiating functions in solving differential equations?
5. How can we verify if multiple functions are solutions to the same equation?

Tip: Always simplify both sides of the equation after substitution to make it easier to compare both sides!