Write An Equation In Slope-intercept Form For The Line That Passesthrough (-8, -32) And Is Perpendicular
Understanding how to find the equation of a line that passes through a specific point and is perpendicular to another line is an essential skill in algebra and coordinate geometry. This comprehensive guide will walk you through the process of deriving the slope-intercept form of such a line, using the point (-8, -32) as an example. Whether you're a student preparing for exams or someone interested in mastering geometric concepts, this article provides step-by-step instructions, key formulas, and practical tips to help you confidently solve similar problems.
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Fundamental Concepts in Line Equations and Perpendicular Lines
Before diving into the specific problem, it's important to review some foundational concepts related to line equations, slopes, and perpendicularity.
What Is the Slope-Intercept Form?
The slope-intercept form of a linear equation is a popular way to express a line's equation:
\[ y = mx + b \]
where:
- \( y \) is the dependent variable,
- \( x \) is the independent variable,
- \( m \) is the slope of the line,
- \( b \) is the y-intercept (the point where the line crosses the y-axis).
This form makes it easy to identify the slope and y-intercept directly from the equation.
Understanding Slope and Its Significance
The slope (\( m \)) indicates the steepness and direction of the line:
- A positive slope means the line inclines upward from left to right.
- A negative slope indicates a downward incline.
- A slope of zero denotes a horizontal line.
- An undefined slope (vertical line) is not expressed in slope-intercept form.
Perpendicular Lines and Their Slopes
Two lines are perpendicular if they intersect at a right angle (90 degrees). The key property relating their slopes is:
\[ m1 \times m2 = -1 \]
where:
- \( m_1 \) is the slope of the first line,
- \( m_2 \) is the slope of the perpendicular line.
This means:
- The slopes are negative reciprocals of each other.
- If one line has slope \( m \), the perpendicular line has slope \( -\frac{1}{m} \).
Understanding this reciprocal relationship is crucial for solving problems involving perpendicular lines.
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Step-by-Step Guide to Find the Equation of the Perpendicular Line
Now, let's proceed with the specific problem: Find the equation in slope-intercept form of the line passing through the point (-8, -32) and perpendicular to another known line or based on a given slope.
Step 1: Determine the Slope of the Original Line
To find a perpendicular line, you need to know the slope of the line to which the new line is perpendicular. Since the problem states "passesthrough (-8, -32) and is perpendicular," but doesn't specify the original line's equation, we can interpret it as:
- Either the original line's slope is given.
- Or, if not given, the problem might be asking to find the line perpendicular to a specific line.
Assumption: If the problem is to find the line perpendicular to a known line with a given slope \( m \), then proceed accordingly.
Example: Suppose the original line has slope \( m_{orig} = \frac{2}{3} \).
In this case, the perpendicular slope \( m_{perp} \) is:
\[ m{perp} = -\frac{1}{m{orig}} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2} \]
Note: If the original line's slope isn't specified, you'll need to know or be provided that value.
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Step 2: Find the Slope of the Perpendicular Line
Using the reciprocal relationship, identify or compute the slope of the perpendicular line:
\[ m{perp} = -\frac{1}{m{orig}} \]
For example, if \( m_{orig} = 4 \), then:
\[ m_{perp} = -\frac{1}{4} \]
If the original line's slope is unknown, and you're asked to find the line perpendicular to a given line, ensure you identify or calculate \( m_{orig} \) first.
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Step 3: Use the Point-Slope Form to Find the Equation
Once the slope of the perpendicular line (\( m_{perp} \)) is known, and the point (-8, -32) is given, use the point-slope form:
\[ y - y1 = m (x - x1) \]
where:
- \( (x1, y1) = (-8, -32) \),
- \( m = m_{perp} \).
Plug in the values:
\[ y - (-32) = m_{perp} (x - (-8)) \]
\[ y + 32 = m_{perp} (x + 8) \]
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Step 4: Convert to Slope-Intercept Form
Distribute \( m_{perp} \):
\[ y + 32 = m{perp} x + 8 m{perp} \]
Subtract 32 from both sides:
\[ y = m{perp} x + 8 m{perp} - 32 \]
This is the slope-intercept form:
\[ y = m_{perp} x + b \]
where the y-intercept \( b = 8 m_{perp} - 32 \).
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Example: Complete Solution with a Specific Original Line Slope
Let's walk through a complete example with specific values.
Suppose the original line has slope \( m_{orig} = 3 \).
Step 1: Find the perpendicular slope:
\[ m_{perp} = -\frac{1}{3} \]
Step 2: Use point (-8, -32) to write the equation:
\[ y - (-32) = -\frac{1}{3} (x - (-8)) \]
\[ y + 32 = -\frac{1}{3} (x + 8) \]
Step 3: Distribute:
\[ y + 32 = -\frac{1}{3} x - \frac{8}{3} \]
Step 4: Subtract 32 from both sides:
\[ y = -\frac{1}{3} x - \frac{8}{3} - 32 \]
Express 32 as a fraction with denominator 3:
\[ 32 = \frac{96}{3} \]
So:
\[ y = -\frac{1}{3} x - \frac{8}{3} - \frac{96}{3} = -\frac{1}{3} x - \frac{104}{3} \]
Final Equation:
\[ y = -\frac{1}{3} x - \frac{104}{3} \]
This is the slope-intercept form of the line passing through (-8, -32) and perpendicular to the original line with slope 3.
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Additional Tips and Common Mistakes
To ensure accuracy and clarity, consider these tips:
- Always identify the original line's slope: The key to finding a perpendicular line is knowing the original slope.
- Convert fractions carefully: When dealing with fractions, ensure proper numerator and denominator handling to avoid mistakes.
- Check your work: Substitute the point into the derived equation to verify correctness.
- Remember vertical and horizontal lines: If the original line is vertical (undefined slope), the perpendicular line will be horizontal (slope 0), and vice versa.
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Practice Problems to Enhance Your Understanding
Try solving these problems to reinforce your skills:
- Find the equation of the line passing through (2, 5) and perpendicular to the line \( y = -\frac{1}{2} x + 3 \).
- Determine the equation of the line passing through (-7, 4) and perpendicular to the line \( y = 4x - 6 \).
- Given the line \( y = 3x + 1 \), find the equation of the line passing through (0, 0) and perpendicular to it.
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Conclusion: Mastering Equations of Perpendicular Lines
Finding the equation in slope-intercept form for a line passing through a specific point and perpendicular to another line involves understanding the relationship between slopes, applying the point-slope formula, and transforming the result into slope-intercept form. By mastering these steps and concepts, you can confidently approach similar problems, enhancing your algebraic and geometric problem-solving skills. Remember, practice makes perfect—so work through various problems to solidify your understanding and become proficient in working with line equations and perpendicularity.
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