To find the equation of a line parallel to $y = 7x + 2$ that passes through the point $(3, -1)$, follow these steps:

Step 1: Identify the slope of the given line

The equation of the given line is in slope-intercept form $y = mx + b$, where:

• $m$ is the slope
• $b$ is the y-intercept

From the equation $y = 7x + 2$, the slope $m$ is $7$.

Step 2: Use the same slope for the parallel line

Since parallel lines have the same slope, the slope of the new line will also be $7$.

Step 3: Use the point-slope form of the equation

The point-slope form of the equation of a line is:

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

Where $m$ is the slope and $(x_1, y_1)$ is a point on the line. Here, the point given is $(3, -1)$, and the slope is $7$.

Substitute into the formula:

$y - (-1) = 7(x - 3)$

Simplify:

$y + 1 = 7(x - 3)$

Distribute the $7$:

$y + 1 = 7x - 21$

Subtract 1 from both sides:

$y = 7x - 22$

Final Answer:

The equation of the line parallel to $y = 7x + 2$ that passes through $(3, -1)$ is:

$y = 7x - 22$

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Would you like a more detailed explanation or any clarifications? Here are some follow-up questions:

1. How would the equation change if the slope was negative?
2. How do you derive the slope of a line from two points?
3. Can you find the equation of a line perpendicular to this one?
4. What are some applications of parallel lines in real life?
5. How do you convert the equation from point-slope form to slope-intercept form?

Tip: To verify if lines are parallel, check if their slopes are the same.