 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, we need two key pieces of information: the slope and a point on the line.

### Step 1: Identify the slope of the given line.
The given line is \( y = 2x + 1 \). This is already in slope-intercept form (\( y = mx + b \)), where \( m \) is the slope of the line.

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

### Step 2: Find the slope of the perpendicular line.
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{1}{2}
\]

### Step 3: Use the point-slope form to find the equation of the line.
The line we're looking for passes through the point \( (5, -3) \) and has a slope of \( -\frac{1}{2} \). We use the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point on the line, and \( m \) is the slope.

Substitute \( (x_1, y_1) = (5, -3) \) and \( m = -\frac{1}{2} \) into the equation:
\[
y - (-3) = -\frac{1}{2}(x - 5)
\]
Simplify:
\[
y + 3 = -\frac{1}{2}(x - 5)
\]

### Step 4: Solve for \( y \) to get the slope-intercept form.
Distribute the slope on the right-hand side:
\[
y + 3 = -\frac{1}{2}x + \frac{5}{2}
\]
Now, subtract 3 from both sides to isolate \( y \):
\[
y = -\frac{1}{2}x + \frac{5}{2} - 3
\]

Convert \( 3 \) to a fraction with denominator 2:
\[
y = -\frac{1}{2}x + \frac{5}{2} - \frac{6}{2}
\]

Now simplify:
\[
y = -\frac{1}{2}x - \frac{1}{2}
\]

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

Would you like more details or have any questions? 

Here are 5 related questions:
1. How do you find the equation of a line given two points?
2. What is the meaning of slope in the context of a linear equation?
3. How can you tell if two lines are parallel or perpendicular based on their equations?
4. How do you convert from point-slope form to slope-intercept form?
5. What happens to the graph of a line when its slope becomes more negative?

**Tip:** Always remember that perpendicular lines have slopes that are negative reciprocals of each other.

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

Learn how to find the equation of a line that passes through the point (5, -3) and is perpendicular to the line y = 2x + 1. This step-by-step guide covers key concepts such as slope-intercept form, point-slope form, and perpendicular slopes. Ideal for students in Grades 8-10.

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