Problem 34
Question
Write the slope-intercept equation of the line that passes through the given point and that is perpendicular to the given line. $$ (0,-5), y=3 x+5 $$
Step-by-Step Solution
Verified Answer
The equation is \( y = -\frac{1}{3}x - 5 \).
1Step 1: Determine the slope of the given line
The given line has the equation \( y = 3x + 5 \). In slope-intercept form, \( y = mx + b \), \( m \) is the slope. Thus, the slope of the given line is \( 3 \).
2Step 2: Find the slope of the perpendicular line
Two lines are perpendicular if the product of their slopes is \(-1\). Therefore, the perpendicular slope \( m' \) can be found using \( 3 imes m' = -1 \). Solving for \( m' \) gives \( m' = -\frac{1}{3} \).
3Step 3: Formulate the slope-intercept equation for the perpendicular line
We already know the slope \( m' = -\frac{1}{3} \) and the line passes through the point \((0, -5)\). Use the point-slope form \( y - y_1 = m(x - x_1) \) where \( (x_1, y_1) = (0, -5) \).
4Step 4: Simplify to slope-intercept form
Substitute \( m = -\frac{1}{3} \), \( x_1 = 0 \), and \( y_1 = -5 \) into the point-slope form: \( y + 5 = -\frac{1}{3}(x - 0) \) which simplifies to \( y = -\frac{1}{3}x - 5 \) when rearranged to \( y = mx + b \).
Key Concepts
Slope-Intercept FormPoint-Slope FormSlope of a Line
Slope-Intercept Form
The slope-intercept form is a method of expressing the equation of a straight line in the form of \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, which indicates where the line crosses the y-axis.
This form is extremely helpful because it allows you to quickly gather key information about the line's slope and its intercept just by looking at the equation.
When dealing with problems involving perpendicular lines, it's important to first identify these two variables, \( m \) and \( b \).
Perpendicular lines have slopes that are negative reciprocals of each other. This means that, if the slope of one line is \( m \), the slope of the line perpendicular to it will be \(-\frac{1}{m}\).
Using this information, once the slope-intercept form of the original line is found, it's easier to calculate the slope of a line perpendicular to it.
This form is extremely helpful because it allows you to quickly gather key information about the line's slope and its intercept just by looking at the equation.
When dealing with problems involving perpendicular lines, it's important to first identify these two variables, \( m \) and \( b \).
Perpendicular lines have slopes that are negative reciprocals of each other. This means that, if the slope of one line is \( m \), the slope of the line perpendicular to it will be \(-\frac{1}{m}\).
Using this information, once the slope-intercept form of the original line is found, it's easier to calculate the slope of a line perpendicular to it.
Point-Slope Form
The point-slope form is a useful formula to find the equation of a line when you know a point through which the line passes and the slope of the line.
The form is given by: \( y - y_1 = m(x - x_1) \).
In this equation, \( m \) is the slope, and \((x_1, y_1)\) is the specific point on the line.
This form allows you to substitute directly the known slope and the coordinates of the given point to find the equation.
The form is given by: \( y - y_1 = m(x - x_1) \).
In this equation, \( m \) is the slope, and \((x_1, y_1)\) is the specific point on the line.
This form allows you to substitute directly the known slope and the coordinates of the given point to find the equation.
- First, plug in the slope you found (like \(-\frac{1}{3}\) for the perpendicular slope).
- Next, insert the point, such as \((0, -5)\), into the equation.
- Simplify the equation to transition smoothly into the slope-intercept form.
Slope of a Line
The slope is a measure of how steep a line is. Mathematically, it's calculated as the "rise over run," which means the change in the y-coordinates divided by the change in the x-coordinates.
This can be expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \) for any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line.
Positive slopes rise to the right, negative slopes fall to the right, and a slope of zero indicates a horizontal line.
For perpendicular lines, it's crucial to remember that their slopes are negative reciprocals. This means if one line has a slope of \(3\), a line perpendicular to it will have a slope of \(-\frac{1}{3}\). This relationship helps in determining the direction and inclination of a line, enhancing comprehension of how lines interact in a plane.
This can be expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \) for any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line.
Positive slopes rise to the right, negative slopes fall to the right, and a slope of zero indicates a horizontal line.
For perpendicular lines, it's crucial to remember that their slopes are negative reciprocals. This means if one line has a slope of \(3\), a line perpendicular to it will have a slope of \(-\frac{1}{3}\). This relationship helps in determining the direction and inclination of a line, enhancing comprehension of how lines interact in a plane.
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