Write an equation of the line through the pointers (a) parallel to the given line (b)perpendicular to the given line
Point(2,1) line 4x-2y=3

To find the equations of the lines through the point \( (2, 1) \) that are either parallel or perpendicular to the given line \( 4x - 2y = 3 \), follow these steps:

### 1. Write the given line in slope-intercept form \( y = mx + b \):
The given line is:
\[
4x - 2y = 3
\]

Solve for \( y \):
\[
-2y = -4x + 3
\]
\[
y = 2x - \frac{3}{2}
\]

So, the slope \( m \) of the given line is \( 2 \).

### 2. Equation of the line parallel to the given line:
Lines that are parallel have the same slope. So, the slope of the parallel line is also \( m = 2 \).

Using the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = 2 \) and \( (x_1, y_1) = (2, 1) \):
\[
y - 1 = 2(x - 2)
\]
Simplify:
\[
y - 1 = 2x - 4
\]
\[
y = 2x - 3
\]

So, the equation of the line parallel to the given line passing through the point \( (2, 1) \) is:
\[
\boxed{y = 2x - 3}
\]

### 3. Equation of the line perpendicular to the given line:
For lines that are perpendicular, the slopes are negative reciprocals. Since the slope of the given line is \( 2 \), the slope of the perpendicular line will be:
\[
m = -\frac{1}{2}
\]

Using the point-slope form again:
\[
y - 1 = -\frac{1}{2}(x - 2)
\]
Simplify:
\[
y - 1 = -\frac{1}{2}x + 1
\]
\[
y = -\frac{1}{2}x + 2
\]

So, the equation of the line perpendicular to the given line passing through the point \( (2, 1) \) is:
\[
\boxed{y = -\frac{1}{2}x + 2}
\]

Would you like more details or have any questions?

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### 5 Relative Questions:
1. How do you find the slope of a line given its equation in standard form?
2. What is the significance of the point-slope form in determining the equation of a line?
3. How do the slopes of parallel and perpendicular lines relate?
4. Can you explain why the slope-intercept form of a line is useful?
5. How would the equations change if the point were different, say \( (3, 2) \)?

### Tip:
Remember that parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals.

Learn how to find equations of lines parallel and perpendicular to the line 4x - 2y = 3 passing through the point (2, 1). Detailed step-by-step solutions provided using point-slope form and slope concepts. Suitable for Grades 9-12.

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