Which Of The Following Is The Equation Of A Line That Passes Through \[$(3, 1)\$\] And Is Parallel To \[$y = \frac{2}{3} X - 3\$\]?A. \[$y = -\frac{3}{2} X + 6\$\]B. \[$y = \frac{2}{3} X - 1\$\]C. \[$y = \frac{2}{3}
Which of the Following is the Equation of a Line that Passes Through (3, 1) and is Parallel to y = 2/3 x - 3?
In mathematics, particularly in the field of geometry, lines are an essential concept. A line is a set of points that extend infinitely in two directions. It can be represented in various forms, including the slope-intercept form, which is the most commonly used form. The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.
Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. In other words, parallel lines have the same slope but different y-intercepts. The equation of a line that is parallel to another line can be found by using the slope of the original line and the coordinates of a point that lies on the new line.
To find the equation of a line that passes through the point (3, 1) and is parallel to the line y = 2/3 x - 3, we need to use the slope-intercept form of a line. The slope of the original line is 2/3, and the point (3, 1) lies on the new line.
Step 1: Find the Slope of the New Line
Since the new line is parallel to the original line, it has the same slope, which is 2/3.
Step 2: Use the Point-Slope Form of a Line
The point-slope form of a line is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point that lies on the line and m is the slope. We can use this form to find the equation of the new line.
Step 3: Substitute the Values into the Point-Slope Form
Substituting the values of the point (3, 1) and the slope 2/3 into the point-slope form, we get:
y - 1 = 2/3(x - 3)
Step 4: Simplify the Equation
To simplify the equation, we can multiply both sides by 3 to eliminate the fraction:
3(y - 1) = 2(x - 3)
Expanding the left-hand side, we get:
3y - 3 = 2x - 6
Adding 3 to both sides, we get:
3y = 2x - 3
Dividing both sides by 3, we get:
y = 2/3 x - 1
Therefore, the equation of a line that passes through (3, 1) and is parallel to y = 2/3 x - 3 is y = 2/3 x - 1.
The correct answer is B. y = 2/3 x - 1.
This problem requires the use of the slope-intercept form and the point-slope form of a line. The student needs to understand the concept of parallel lines and how to find the equation of a line that passes through a given point and is parallel to another line. The student also needs to be able to simplify the equation and express it in the slope-intercept form.
- Make sure to use the correct form of a line, either the slope-intercept form or the point-slope form.
- Use the slope of the original line to find the slope of the new line.
- Substitute the values of the point and the slope into the point-slope form.
- Simplify the equation by multiplying both sides by a common factor or by adding or subtracting the same value from both sides.
- Express the equation in the slope-intercept form by dividing both sides by the coefficient of x.
- Find the equation of a line that passes through (2, 4) and is parallel to y = 1/2 x + 2.
- Find the equation of a line that passes through (1, 3) and is parallel to y = 3/4 x - 1.
- Find the equation of a line that passes through (4, 2) and is parallel to y = 2/5 x + 3.
- y = 1/2 x + 4
- y = 3/4 x + 2
- y = 2/5 x + 1
Q&A: Lines and Equations ==========================
Q: What is the equation of a line that passes through (2, 3) and is parallel to y = 2x - 1?
A: To find the equation of a line that passes through (2, 3) and is parallel to y = 2x - 1, we need to use the slope-intercept form of a line. The slope of the original line is 2, and the point (2, 3) lies on the new line. Using the point-slope form, we get:
y - 3 = 2(x - 2)
Simplifying the equation, we get:
y - 3 = 2x - 4
Adding 3 to both sides, we get:
y = 2x - 1
Therefore, the equation of a line that passes through (2, 3) and is parallel to y = 2x - 1 is y = 2x - 1.
Q: What is the equation of a line that passes through (4, 2) and is perpendicular to y = 3x + 1?
A: To find the equation of a line that passes through (4, 2) and is perpendicular to y = 3x + 1, we need to use the slope-intercept form of a line. The slope of the original line is 3, and the point (4, 2) lies on the new line. Since the new line is perpendicular to the original line, its slope is the negative reciprocal of the original slope, which is -1/3.
Using the point-slope form, we get:
y - 2 = -1/3(x - 4)
Simplifying the equation, we get:
y - 2 = -1/3x + 4/3
Adding 2 to both sides, we get:
y = -1/3x + 10/3
Therefore, the equation of a line that passes through (4, 2) and is perpendicular to y = 3x + 1 is y = -1/3x + 10/3.
Q: What is the equation of a line that passes through (1, 2) and is parallel to y = x + 2?
A: To find the equation of a line that passes through (1, 2) and is parallel to y = x + 2, we need to use the slope-intercept form of a line. The slope of the original line is 1, and the point (1, 2) lies on the new line. Using the point-slope form, we get:
y - 2 = 1(x - 1)
Simplifying the equation, we get:
y - 2 = x - 1
Adding 2 to both sides, we get:
y = x + 1
Therefore, the equation of a line that passes through (1, 2) and is parallel to y = x + 2 is y = x + 1.
Q: What is the equation of a line that passes through (3, 4) and is perpendicular to y = 2x - 3?
A: To find the equation of a line that passes through (3, 4) and is perpendicular to y = 2x - 3, we need to use the slope-intercept form of a line. The slope of the original line is 2, and the point (3, 4) lies on the new line. Since the new line is perpendicular to the original line, its slope is the negative reciprocal of the original slope, which is -1/2.
Using the point-slope form, we get:
y - 4 = -1/2(x - 3)
Simplifying the equation, we get:
y - 4 = -1/2x + 3/2
Adding 4 to both sides, we get:
y = -1/2x + 11/2
Therefore, the equation of a line that passes through (3, 4) and is perpendicular to y = 2x - 3 is y = -1/2x + 11/2.
Q: What is the equation of a line that passes through (2, 1) and is parallel to y = x - 1?
A: To find the equation of a line that passes through (2, 1) and is parallel to y = x - 1, we need to use the slope-intercept form of a line. The slope of the original line is 1, and the point (2, 1) lies on the new line. Using the point-slope form, we get:
y - 1 = 1(x - 2)
Simplifying the equation, we get:
y - 1 = x - 2
Adding 1 to both sides, we get:
y = x - 1
Therefore, the equation of a line that passes through (2, 1) and is parallel to y = x - 1 is y = x - 1.
Q: What is the equation of a line that passes through (4, 3) and is perpendicular to y = x + 2?
A: To find the equation of a line that passes through (4, 3) and is perpendicular to y = x + 2, we need to use the slope-intercept form of a line. The slope of the original line is 1, and the point (4, 3) lies on the new line. Since the new line is perpendicular to the original line, its slope is the negative reciprocal of the original slope, which is -1.
Using the point-slope form, we get:
y - 3 = -1(x - 4)
Simplifying the equation, we get:
y - 3 = -x + 4
Adding 3 to both sides, we get:
y = -x + 7
Therefore, the equation of a line that passes through (4, 3) and is perpendicular to y = x + 2 is y = -x + 7.
In this article, we have discussed various questions related to lines and equations. We have used the slope-intercept form and the point-slope form of a line to find the equations of lines that pass through given points and are parallel or perpendicular to other lines. We have also provided solutions to the questions and explained the concepts behind the solutions.