Problem 17
Question
Use the facts that parallel lines have equal slopes and that the slopes of perpendicular lines are negative reciprocals of one another. Find an equation for the line through the point (2,1) which is perpendicular to the line \(y=5 x-3\)
Step-by-Step Solution
Verified Answer
The equation is \( y = -\frac{1}{5}x + \frac{7}{5} \).
1Step 1: Identify the Slope of the Given Line
The given line is in the form \( y = mx + b \) where \( m \) is the slope. From the equation \( y = 5x - 3 \), the slope \( m \) is 5.
2Step 2: Find the Slope of the Perpendicular Line
Since perpendicular lines have slopes that are negative reciprocals, the slope of the line perpendicular to \( y = 5x - 3 \) is \( -\frac{1}{5} \). This is because the negative reciprocal of 5 is \( -\frac{1}{5} \).
3Step 3: Write the Equation of the Perpendicular Line
Use the point-slope form: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the point on the line. Plug in the point (2, 1) and the slope \( -\frac{1}{5} \): \( y - 1 = -\frac{1}{5}(x - 2) \).
4Step 4: Simplify the Equation
Distribute the slope and simplify: \( y - 1 = -\frac{1}{5}x + \frac{2}{5} \). Add 1 to both sides to solve for \( y \): \( y = -\frac{1}{5}x + \frac{2}{5} + 1 \). Simplify further: \( y = -\frac{1}{5}x + \frac{7}{5} \). The equation of the line is \( y = -\frac{1}{5}x + \frac{7}{5} \).
Key Concepts
Slope of a LinePoint-Slope FormNegative Reciprocal
Slope of a Line
The slope of a line is a measure that describes the steepness and direction of the line. It is usually represented by the letter \( m \) in the equation of a line. When a line is expressed in the form \( y = mx + b \), \( m \) specifies how much \( y \) changes for every unit of change in \( x \).
For the line \( y = 5x - 3 \), the slope \( m \) is 5. This means for every increase of 1 in \( x \), \( y \) increases by 5. A positive slope like 5 indicates the line rises from left to right. Conversely, a negative slope means the line falls. Understanding the slope is crucial to solving problems involving parallel and perpendicular lines.
The slope provides key insights into the behavior of a line, helping determine if two lines are parallel (identical slopes) or perpendicular (negative reciprocal slopes). Identifying the slope is the first step in solving geometry problems involving lines.
For the line \( y = 5x - 3 \), the slope \( m \) is 5. This means for every increase of 1 in \( x \), \( y \) increases by 5. A positive slope like 5 indicates the line rises from left to right. Conversely, a negative slope means the line falls. Understanding the slope is crucial to solving problems involving parallel and perpendicular lines.
The slope provides key insights into the behavior of a line, helping determine if two lines are parallel (identical slopes) or perpendicular (negative reciprocal slopes). Identifying the slope is the first step in solving geometry problems involving lines.
Point-Slope Form
The point-slope form of the equation of a line is a useful way to write the equation when a point on the line and the slope are known. The formula is \( y - y_1 = m(x - x_1) \).
This formula helps us quickly draft the equation of a line using a slope \( m \) and a point \((x_1, y_1)\).
In our exercise, to find the equation for the line through the point (2, 1) that is perpendicular to the line \( y = 5x - 3 \), we first identified the new slope by taking the negative reciprocal of the given slope (5), resulting in \(-\frac{1}{5}\). With this slope and the point (2, 1), we apply the point-slope form: \( y - 1 = -\frac{1}{5}(x - 2) \).
Using this form is particularly efficient when a perpendicular or parallel line needs to be established, as it directly incorporates the critical slope and point information.
This formula helps us quickly draft the equation of a line using a slope \( m \) and a point \((x_1, y_1)\).
In our exercise, to find the equation for the line through the point (2, 1) that is perpendicular to the line \( y = 5x - 3 \), we first identified the new slope by taking the negative reciprocal of the given slope (5), resulting in \(-\frac{1}{5}\). With this slope and the point (2, 1), we apply the point-slope form: \( y - 1 = -\frac{1}{5}(x - 2) \).
Using this form is particularly efficient when a perpendicular or parallel line needs to be established, as it directly incorporates the critical slope and point information.
Negative Reciprocal
The concept of negative reciprocal is fundamental when dealing with perpendicular lines. It states that if two lines are perpendicular, the slopes of these lines multiply together to give \(-1\). This means the slope of one line is the negative reciprocal of the other.
For example, if a line has a slope of 5, like \( y = 5x - 3 \), the slope of a line perpendicular to it will be \(-\frac{1}{5}\). Let's break this down: the reciprocal of 5 is \( \frac{1}{5} \), and making it negative gives us \(-\frac{1}{5}\).
Applying negative reciprocal understanding allows us to find perpendicular lines easily and effectively. It's especially handy in geometric problem-solving, ensuring quick identification of perpendicular relationships by simply adjusting the slope to its negative reciprocal counterpart.
For example, if a line has a slope of 5, like \( y = 5x - 3 \), the slope of a line perpendicular to it will be \(-\frac{1}{5}\). Let's break this down: the reciprocal of 5 is \( \frac{1}{5} \), and making it negative gives us \(-\frac{1}{5}\).
Applying negative reciprocal understanding allows us to find perpendicular lines easily and effectively. It's especially handy in geometric problem-solving, ensuring quick identification of perpendicular relationships by simply adjusting the slope to its negative reciprocal counterpart.
Other exercises in this chapter
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