To find the equation of the line containing the point $(-2, 2)$ and parallel to the given 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 into slope-intercept form.

$8x = 9y + 5$

First, solve for $y$:

$9y = 8x - 5$

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

So, the slope $m$ of the given line is $\frac{8}{9}$.

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

Since the lines are parallel, the new line will have the same slope $m = \frac{8}{9}$. Use the point-slope form of a line equation:

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

Where:

• $m = \frac{8}{9}$
• $(x_1, y_1) = (-2, 2)$

Substitute these values into the equation:

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

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 (which is $\frac{18}{9}$ in terms of ninths) to both sides to solve for $y$:

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

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

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 further details or have any questions?

Here are some related questions:

1. How do you find the equation of a line perpendicular to a given line?
2. What is the significance of the slope in parallel lines?
3. How do you convert a line equation from standard form to slope-intercept form?
4. Can you find the equation of a line passing through two points?
5. How do you interpret the y-intercept in a linear equation?

Tip: Always check if two lines are parallel by comparing their slopes. Parallel lines have the same slope.