To find the equation of the line that passes through the point $(-2, 2)$ and is parallel to the line $8x = 9y + 5$, follow these steps:

Step 1: Rewrite the given line in slope-intercept form ($y = mx + b$)

Start by converting the given equation $8x = 9y + 5$ into slope-intercept form:

$8x = 9y + 5$

Subtract 5 from both sides to isolate the $9y$ term:

$8x - 5 = 9y$

Now, divide both sides by 9 to solve for $y$:

$y = \frac{8}{9}x - \frac{5}{9}$

This gives us the slope $m = \frac{8}{9}$.

Step 2: Use the point-slope form to find the equation of the new line

Since the new line is parallel to the given line, it will have the same slope $m = \frac{8}{9}$. We can use the point-slope form of the line equation, which is:

$y - y_1 = m(x - x_1)$

Where $(x_1, y_1) = (-2, 2)$. Substitute the values:

$y - 2 = \frac{8}{9}(x + 2)$

Step 3: Simplify to slope-intercept form ($y = mx + b$)

Distribute the slope $\frac{8}{9}$ on the right side:

$y - 2 = \frac{8}{9}x + \frac{8}{9} \times 2$

$y - 2 = \frac{8}{9}x + \frac{16}{9}$

Now, add 2 to both sides to solve for $y$ (remember that 2 is equivalent to $\frac{18}{9}$ in terms of ninths):

$y = \frac{8}{9}x + \frac{16}{9} + \frac{18}{9}$

$y = \frac{8}{9}x + \frac{34}{9}$

Final Answer:

The equation of the line in the form $y = mx + b$ is:

$y = \frac{8}{9}x + \frac{34}{9}$

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the slope of a line given two points?
2. How can you determine if two lines are perpendicular?
3. What is the point-slope form of a linear equation?
4. How do you convert a linear equation from slope-intercept form to standard form?
5. What is the significance of the y-intercept in a linear equation?

Tip: When given a problem involving parallel lines, remember that they always share the same slope.