What Is The Equation Of A Line That Contains The Point { (2, -5)$}$ And Is Parallel To The Line { Y = 3x - 4$}$?A. { Y = -\frac{1}{3}x - 4$}$B. { Y = -\frac{1}{3}x - \frac{13}{3}$}$C. { Y = 3x - 11$}$D.
What is the Equation of a Line that Contains the Point (2, -5) and is Parallel to the Line y = 3x - 4?
Understanding the Basics of Linear Equations
In mathematics, a linear equation is a type of equation that represents a straight line on a graph. The general form of a linear equation is y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is a measure of how steep it is, and it can be calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Parallel Lines and Slope
Two lines are said to be parallel if they have the same slope but different y-intercepts. In other words, if two lines have the same value of m, they are parallel. This is because the slope of a line determines its steepness, and if two lines have the same steepness, they will never intersect.
The Given Line and Its Slope
The given line is y = 3x - 4. To find its slope, we can rewrite it in the form y = mx + b. In this case, the slope m is 3, and the y-intercept b is -4. This means that the given line has a slope of 3.
The Point (2, -5) and Its Significance
The point (2, -5) lies on the line we are trying to find. This means that the line we are looking for must pass through this point. We can use this point to help us find the equation of the line.
Finding the Equation of the Line
Since the line we are looking for is parallel to the given line, it must have the same slope as the given line. Therefore, the slope of the line we are looking for is also 3. Now, we can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Plugging in the Values
We know that the point (2, -5) lies on the line, so we can plug in x1 = 2 and y1 = -5. We also know that the slope m is 3. Plugging these values into the point-slope form, we get:
y - (-5) = 3(x - 2)
Simplifying the Equation
To simplify the equation, we can start by distributing the 3 to the terms inside the parentheses:
y + 5 = 3x - 6
Isolating y
Now, we can isolate y by subtracting 5 from both sides of the equation:
y = 3x - 11
Conclusion
Therefore, the equation of the line that contains the point (2, -5) and is parallel to the line y = 3x - 4 is y = 3x - 11.
Answer
The correct answer is C. y = 3x - 11.
Discussion
This problem requires the application of the concept of parallel lines and the use of the point-slope form of a linear equation. The student must understand that two lines are parallel if they have the same slope but different y-intercepts, and that the point-slope form of a linear equation can be used to find the equation of a line that passes through a given point and has a given slope.
Key Concepts
- Parallel lines
- Slope
- Point-slope form of a linear equation
- Linear equations
Mathematical Operations
- Distributing
- Isolating variables
Problem-Solving Strategies
- Using the point-slope form of a linear equation
- Applying the concept of parallel lines
Real-World Applications
- Finding the equation of a line that passes through a given point and has a given slope
- Understanding the concept of parallel lines in real-world situations
Extension
- Finding the equation of a line that passes through two given points
- Understanding the concept of perpendicular lines
Common Mistakes
- Failing to recognize that the line we are looking for is parallel to the given line
- Failing to use the point-slope form of a linear equation to find the equation of the line
Tips for Success
- Make sure to understand the concept of parallel lines and the point-slope form of a linear equation
- Use the point-slope form of a linear equation to find the equation of the line
- Check your work by plugging in the values of the point and the slope into the equation.
Q&A: What is the Equation of a Line that Contains the Point (2, -5) and is Parallel to the Line y = 3x - 4?
Frequently Asked Questions
Q: What is the equation of a line that contains the point (2, -5) and is parallel to the line y = 3x - 4?
A: The equation of the line that contains the point (2, -5) and is parallel to the line y = 3x - 4 is y = 3x - 11.
Q: Why is the line we are looking for parallel to the given line?
A: The line we are looking for is parallel to the given line because they have the same slope. The slope of the given line is 3, and the slope of the line we are looking for is also 3.
Q: How do we find the equation of the line that contains the point (2, -5) and is parallel to the line y = 3x - 4?
A: We can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: How do we use the point-slope form to find the equation of the line?
A: We can plug in the values of the point and the slope into the point-slope form to find the equation of the line. In this case, we know that the point (2, -5) lies on the line, and the slope is 3.
Q: What is the equation of the line that we found using the point-slope form?
A: The equation of the line that we found using the point-slope form is y = 3x - 11.
Q: Why is the equation of the line y = 3x - 11 correct?
A: The equation of the line y = 3x - 11 is correct because it satisfies the conditions of the problem. The line contains the point (2, -5) and is parallel to the line y = 3x - 4.
Q: What are some common mistakes that students make when solving this type of problem?
A: Some common mistakes that students make when solving this type of problem include:
- Failing to recognize that the line we are looking for is parallel to the given line
- Failing to use the point-slope form of a linear equation to find the equation of the line
- Failing to check their work by plugging in the values of the point and the slope into the equation.
Q: How can students avoid making these mistakes?
A: Students can avoid making these mistakes by:
- Making sure to understand the concept of parallel lines and the point-slope form of a linear equation
- Using the point-slope form of a linear equation to find the equation of the line
- Checking their work by plugging in the values of the point and the slope into the equation.
Q: What are some real-world applications of the concept of parallel lines and the point-slope form of a linear equation?
A: Some real-world applications of the concept of parallel lines and the point-slope form of a linear equation include:
- Finding the equation of a line that passes through a given point and has a given slope
- Understanding the concept of parallel lines in real-world situations, such as in architecture or engineering.
Q: How can students apply the concept of parallel lines and the point-slope form of a linear equation to real-world problems?
A: Students can apply the concept of parallel lines and the point-slope form of a linear equation to real-world problems by:
- Using the point-slope form of a linear equation to find the equation of a line that passes through a given point and has a given slope
- Understanding the concept of parallel lines in real-world situations and applying it to solve problems.
Q: What are some tips for success when solving problems involving parallel lines and the point-slope form of a linear equation?
A: Some tips for success when solving problems involving parallel lines and the point-slope form of a linear equation include:
- Making sure to understand the concept of parallel lines and the point-slope form of a linear equation
- Using the point-slope form of a linear equation to find the equation of the line
- Checking your work by plugging in the values of the point and the slope into the equation.