Let's solve this step-by-step.

1. Identify the given line’s slope

The equation you provided is:

$y = x + 4x + 2 \quad \text{or} \quad y = 5x + 2$

Thus, the slope of this line is 5.

2. Write the equation of the parallel line

Since the new line is parallel to the given one, it must have the same slope, which is $m = 5$. Therefore, the general form of the parallel line’s equation is:

$y = 5x + b$

We need to determine the value of $b$, the y-intercept, to ensure the line passes through the point $(2, 2)$.

3. Substitute the point $(2, 2)$

We substitute $x = 2$ and $y = 2$ into the equation:

$2 = 5(2) + b$

$2 = 10 + b$

$b = 2 - 10 = -8$

4. Write the final equation

The equation of the line that passes through $(2, 2)$ and is parallel to $y = 5x + 2$ is:

$y = 5x - 8$

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Let me know if any step needs further explanation!

Related Questions:

1. How do you find the slope of a line from its equation?
2. What does it mean for two lines to be parallel?
3. How do you determine the equation of a line from a point and a slope?
4. How do you write the equation of a line in point-slope form?
5. Can you explain how to convert from point-slope form to slope-intercept form?

Tip: Parallel lines always have the same slope, but different y-intercepts unless they are identical lines!