Find an equation of the line​ L, where L is perpendicular to y equals five halves xy=
5
2x and passes through the point ​(2​,5​).

To find the equation of the line \( L \), which is perpendicular to the line \( y = \frac{5}{2}x \) and passes through the point \( (2, 5) \), we need to follow these steps:

### Step 1: Determine the slope of the given line
The given line is \( y = \frac{5}{2}x \). The slope of this line, \( m_1 \), is the coefficient of \( x \), which is:
\[
m_1 = \frac{5}{2}
\]

### Step 2: Find the slope of the perpendicular line
Two lines are perpendicular if the product of their slopes is \( -1 \). Let the slope of the line \( L \) be \( m_2 \). The relationship between the slopes is:
\[
m_1 \cdot m_2 = -1
\]
Substituting \( m_1 = \frac{5}{2} \):
\[
\frac{5}{2} \cdot m_2 = -1
\]
Solving for \( m_2 \):
\[
m_2 = -\frac{2}{5}
\]

### Step 3: Use the point-slope form to write the equation of the line
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Here, \( m = -\frac{2}{5} \) and the point is \( (2, 5) \). Substituting these values:
\[
y - 5 = -\frac{2}{5}(x - 2)
\]

### Step 4: Simplify the equation
Now, expand the right-hand side:
\[
y - 5 = -\frac{2}{5}x + \frac{4}{5}
\]
Add 5 to both sides:
\[
y = -\frac{2}{5}x + \frac{4}{5} + 5
\]
Convert 5 to a fraction:
\[
y = -\frac{2}{5}x + \frac{4}{5} + \frac{25}{5}
\]
Simplify the constants:
\[
y = -\frac{2}{5}x + \frac{29}{5}
\]

### Final equation:
The equation of the line \( L \) is:
\[
y = -\frac{2}{5}x + \frac{29}{5}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you find the slope of a line given two points?
2. What is the general equation for a line in slope-intercept form?
3. How do you verify if two lines are perpendicular?
4. Can you rewrite the equation in standard form?
5. How does the point-slope form differ from the slope-intercept form?

**Tip:** When dealing with perpendicular lines, always remember that their slopes are negative reciprocals of each other!

Find an equation of the line L, where L is perpendicular to y = 5/2x and passes through the point (2,5).

Discover how to find the equation of a line that is perpendicular to y = 5/2x and passes through the point (2, 5). This guide explores key concepts such as slope-intercept form, perpendicular slopes, and point-slope form, providing a detailed step-by-step solution. Suitable for high school students in Grades 9-12.

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