Find the slope-intercept form of the equation of the line that passes through the point (5, -3) and perpendicular to line y = 2x + 1

To find the slope-intercept form of the equation of a line that passes through the point \((5, -3)\) and is perpendicular to the line \(y = 2x + 1\), follow these steps:

### 1. Identify the slope of the given line
The given line is \(y = 2x + 1\). The slope of this line, \(m_1\), is 2.

### 2. Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, if the slope of the given line is 2, the slope \(m_2\) of the perpendicular line is:
\[
m_2 = -\frac{1}{2}
\]

### 3. Use the point-slope form of the equation of a line
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Substituting the slope \(m_2 = -\frac{1}{2}\) and the point \((x_1, y_1) = (5, -3)\):
\[
y + 3 = -\frac{1}{2}(x - 5)
\]

### 4. Simplify to get the slope-intercept form
Expand and simplify the equation to convert it to the slope-intercept form \(y = mx + b\):
\[
y + 3 = -\frac{1}{2}x + \frac{5}{2}
\]
Subtract 3 from both sides:
\[
y = -\frac{1}{2}x + \frac{5}{2} - 3
\]
Convert \(-3\) to a fraction:
\[
y = -\frac{1}{2}x + \frac{5}{2} - \frac{6}{2}
\]
Combine the fractions:
\[
y = -\frac{1}{2}x - \frac{1}{2}
\]

### Final Answer
The equation of the line in slope-intercept form is:
\[
y = -\frac{1}{2}x - \frac{1}{2}
\]

Would you like further details on this process or have any questions?

Here are five related questions to consider:
1. What is the equation of a line parallel to \(y = 2x + 1\) passing through the point \((5, -3)\)?
2. How do you find the equation of a line given two points?
3. What is the significance of the slope-intercept form of a line?
4. How do you calculate the distance between a point and a line?
5. What are the properties of perpendicular lines in geometry?

**Tip:** Remember that perpendicular slopes are negative reciprocals, a useful concept in coordinate geometry.

Learn how to find the slope-intercept form of a line that passes through the point (5, -3) and is perpendicular to the line y = 2x + 1. This problem involves understanding perpendicular slopes and applying the point-slope form of a line. Suitable for Grades 7-9 students studying coordinate geometry.

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