Problem 36
Question
The equation of a line is given. Find the slope of a line that is a. parallel to the line with the given equation; and b. perpendicular to the line with the given equation. $$y=-\frac{3}{7} x-2$$
Step-by-Step Solution
Verified Answer
Therefore, the slope of a line parallel to the given line is -3/7 and the slope of a line perpendicular to the given line is 7/3.
1Step 1: Finding the Slope of a Parallel Line
Observe the equation of the line: \( y = -\frac{3}{7}x - 2 \). This is in the slope-intercept form of a line equation \( y = mx + b \) where \( m \) is the slope, and \( b \) is the y-intercept of the line. So, the slope of this given line is -3/7. Since parallel lines have the same slope, the slope of the line parallel to the given line is also -3/7.
2Step 2: Finding the Slope of a Perpendicular Line
For a line to be perpendicular to another, its slope is the negative reciprocal of the slope of the given line. The negative reciprocal is obtained by reversing the fraction and changing the sign. For a slope of -3/7, the negative reciprocal is found by reversing the fraction to get 7/3 and then changing the sign to get -7/3. So the slope of a line perpendicular to this given line is 7/3.
Key Concepts
Parallel and Perpendicular LinesSlope-Intercept FormThe Negative Reciprocal
Parallel and Perpendicular Lines
Understanding the relationship between parallel and perpendicular lines is crucial in geometry, and it ties directly to their slopes when we consider them in a coordinate plane.
For lines to be parallel, they must run alongside each other at a constant distance, never meeting no matter how far they are extended. In the context of their slopes, this means that parallel lines have identical slopes. If you're given the equation of a line like \( y = -\frac{3}{7}x - 2 \), and you need to find the slope of a line parallel to it, the solution is straightforward: since the slope (\( m \)) is \( -\frac{3}{7} \), the slope of any line parallel to the original will also be \( -\frac{3}{7} \).
The concept of lines being perpendicular to each other introduces a contrasting relation. Perpendicular lines intersect at a right angle (90 degrees). When it comes to their slopes, if one line has a slope of \( m \), the line perpendicular to it will have a slope of the negative reciprocal of \( m \). The negative reciprocal involves flipping the fraction and changing the sign. For example, the negative reciprocal of \( -\frac{3}{7} \) is \( \frac{7}{3} \), which is the slope of a perpendicular line to the original one.
Thus, the visualization and calculation of slopes play a foundational role in determining how lines will interact on the coordinate plane.
For lines to be parallel, they must run alongside each other at a constant distance, never meeting no matter how far they are extended. In the context of their slopes, this means that parallel lines have identical slopes. If you're given the equation of a line like \( y = -\frac{3}{7}x - 2 \), and you need to find the slope of a line parallel to it, the solution is straightforward: since the slope (\( m \)) is \( -\frac{3}{7} \), the slope of any line parallel to the original will also be \( -\frac{3}{7} \).
The concept of lines being perpendicular to each other introduces a contrasting relation. Perpendicular lines intersect at a right angle (90 degrees). When it comes to their slopes, if one line has a slope of \( m \), the line perpendicular to it will have a slope of the negative reciprocal of \( m \). The negative reciprocal involves flipping the fraction and changing the sign. For example, the negative reciprocal of \( -\frac{3}{7} \) is \( \frac{7}{3} \), which is the slope of a perpendicular line to the original one.
Thus, the visualization and calculation of slopes play a foundational role in determining how lines will interact on the coordinate plane.
Slope-Intercept Form
The slope-intercept form of a line is an efficient way to understand and graph linear equations. It's presented as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) indicates the y-intercept, which is where the line crosses the y-axis.
In the given equation \( y = -\frac{3}{7}x - 2 \), the slope-intercept form readily shows us that the slope is \( -\frac{3}{7} \), and the y-intercept is -2. The slope dictates the direction and steepness of the line, while the y-intercept gives a starting point for plotting the line. To draw the line, you'd start at the y-intercept on the y-axis and then use the slope to find another point. For every move of 7 units to the right (positive direction on the x-axis), you'd move 3 units down (negative direction on the y-axis) due to the negative slope.
With this understanding, you can graph any linear equation quickly, identify the rate of change, and make predictions about the values within the graphed line.
In the given equation \( y = -\frac{3}{7}x - 2 \), the slope-intercept form readily shows us that the slope is \( -\frac{3}{7} \), and the y-intercept is -2. The slope dictates the direction and steepness of the line, while the y-intercept gives a starting point for plotting the line. To draw the line, you'd start at the y-intercept on the y-axis and then use the slope to find another point. For every move of 7 units to the right (positive direction on the x-axis), you'd move 3 units down (negative direction on the y-axis) due to the negative slope.
With this understanding, you can graph any linear equation quickly, identify the rate of change, and make predictions about the values within the graphed line.
The Negative Reciprocal
The term 'negative reciprocal' might sound complex, but it's actually a simple and powerful concept used to find the slope of a line perpendicular to another.
If you have a slope, \( m \), the negative reciprocal is found by inverting the fraction and then changing the sign. This mathematical maneuver allows us to ensure that the resulting two lines will meet at right angles. For instance, if a slope is a positive fraction like \( \frac{2}{5} \), the negative reciprocal would be \( -\frac{5}{2} \). Conversely, a negative slope such as \( -\frac{3}{7} \) becomes \( \frac{7}{3} \) when we apply the negative reciprocal.
By using the negative reciprocal, you hinge on the fact that the product of the slopes of two perpendicular lines is -1 (as \( m \times -\frac{1}{m} = -1 \)). It's a quick tool thatGeometry and algebra students must grasp to solve problems involving perpendicular lines in the coordinate system.
If you have a slope, \( m \), the negative reciprocal is found by inverting the fraction and then changing the sign. This mathematical maneuver allows us to ensure that the resulting two lines will meet at right angles. For instance, if a slope is a positive fraction like \( \frac{2}{5} \), the negative reciprocal would be \( -\frac{5}{2} \). Conversely, a negative slope such as \( -\frac{3}{7} \) becomes \( \frac{7}{3} \) when we apply the negative reciprocal.
By using the negative reciprocal, you hinge on the fact that the product of the slopes of two perpendicular lines is -1 (as \( m \times -\frac{1}{m} = -1 \)). It's a quick tool thatGeometry and algebra students must grasp to solve problems involving perpendicular lines in the coordinate system.
Other exercises in this chapter
Problem 35
Use intercepts and a checkpoint to graph each equation. $$x+2 y=0$$
View solution Problem 35
In which quadrants do the \(x\) -coordinates and the \(y\) -coordinates have the same sign?
View solution Problem 36
In Exercises \(27-38,\) graph each linear equation using the slope and y-intercept $$y=-\frac{2}{3} x+4$$
View solution Problem 36
Determine whether the lines through each pair of points are parallel, perpendicular, or neither. $$(-2,-15)\( and \)(0,-3) ;(-12,6)\( and \)(6,3)$$
View solution