find the equations for the lines through P that are (a) parallel and (b) perpendicular to the given line P(2, 1) y=x+2

We are given the point \( P(2, 1) \) and the line equation \( y = x + 2 \), and we need to find the equations of lines through \( P \) that are:

### (a) Parallel to the given line
The slope of the given line \( y = x + 2 \) is 1, since the equation is in the slope-intercept form \( y = mx + b \), where \( m = 1 \).

For the parallel line, the slope must be the same. Thus, the slope of the line passing through \( P(2, 1) \) and parallel to \( y = x + 2 \) is also 1. The equation of a line can be written in point-slope form:

\[
y - y_1 = m(x - x_1)
\]

where \( m \) is the slope and \( (x_1, y_1) \) is the point the line passes through. Substituting \( m = 1 \) and \( (x_1, y_1) = (2, 1) \):

\[
y - 1 = 1(x - 2)
\]

Simplifying:

\[
y - 1 = x - 2
\]
\[
y = x - 1
\]

Thus, the equation of the line parallel to \( y = x + 2 \) and passing through \( P(2, 1) \) is:

\[
y = x - 1
\]

### (b) Perpendicular to the given line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Since the slope of \( y = x + 2 \) is 1, the slope of the perpendicular line will be \( -1 \).

Using the point-slope form again, with \( m = -1 \) and \( (x_1, y_1) = (2, 1) \):

\[
y - 1 = -1(x - 2)
\]

Simplifying:

\[
y - 1 = -x + 2
\]
\[
y = -x + 3
\]

Thus, the equation of the line perpendicular to \( y = x + 2 \) and passing through \( P(2, 1) \) is:

\[
y = -x + 3
\]

### Final Answers:
- (a) Parallel line: \( y = x - 1 \)
- (b) Perpendicular line: \( y = -x + 3 \)

Do you need any more details or have questions about this?

### Related Questions:
1. How do you find the slope of a line from two points?
2. What is the relationship between slopes of parallel lines?
3. How can you derive the point-slope form from the slope-intercept form?
4. What is the general formula for the slope of a perpendicular line?
5. How do we find the distance between a point and a line?

### Tip:
Always check the slope when dealing with parallel and perpendicular lines—parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals.

Learn how to find the equations of lines parallel and perpendicular to the line y = x + 2 passing through the point P(2,1). This solution covers key concepts in algebra and geometry, including point-slope form and properties of parallel and perpendicular lines. Suitable for Grades 8-10.

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