Which Equation Represents A Line That Passes Through { (-9,-3)$}$ And Has A Slope Of { -6$}$?A. { Y - 9 = -6(x - 3)$}$B. { Y + 9 = -6(x + 3)$}$C. { Y - 3 = -6(x - 9)$}$D. { Y + 3 = -6(x + 9)$}$
Understanding the Problem
To find the equation of a line that passes through a given point and has a specific slope, we need to use the point-slope form of a linear equation. The point-slope form is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line.
Given Information
We are given a point (-9, -3) and a slope of -6. We need to find the equation of the line that passes through this point and has this slope.
Using the Point-Slope Form
To find the equation of the line, we can plug in the given point and slope into the point-slope form:
y - (-3) = -6(x - (-9))
Simplifying the Equation
Now, let's simplify the equation:
y + 3 = -6(x + 9)
Comparing with the Options
We can compare this equation with the given options:
A. y - 9 = -6(x - 3) B. y + 9 = -6(x + 3) C. y - 3 = -6(x - 9) D. y + 3 = -6(x + 9)
Conclusion
Based on the comparison, we can see that the equation we derived matches option D. Therefore, the correct answer is:
D. y + 3 = -6(x + 9)
Why This Equation Represents the Line
This equation represents the line because it passes through the given point (-9, -3) and has a slope of -6. The equation is in the point-slope form, which ensures that the line passes through the given point and has the specified slope.
How to Use This Equation
To use this equation, you can plug in any x-value to find the corresponding y-value. For example, if you want to find the y-value when x = 0, you can plug in x = 0 into the equation:
y + 3 = -6(0 + 9) y + 3 = -54 y = -57
Therefore, the y-value when x = 0 is -57.
Real-World Applications
This equation has many real-world applications, such as:
- Physics: The equation can be used to describe the motion of an object under constant acceleration.
- Engineering: The equation can be used to design and optimize systems, such as bridges and buildings.
- Computer Science: The equation can be used to develop algorithms and models for machine learning and data analysis.
Conclusion
In conclusion, the equation that represents a line that passes through (-9, -3) and has a slope of -6 is:
D. y + 3 = -6(x + 9)
This equation is in the point-slope form, which ensures that the line passes through the given point and has the specified slope. The equation has many real-world applications and can be used to solve problems in physics, engineering, and computer science.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line.
Q: How do I find the equation of a line that passes through a given point and has a specific slope?
A: To find the equation of a line that passes through a given point and has a specific slope, you can use the point-slope form of a linear equation. Simply plug in the given point and slope into the equation:
y - y1 = m(x - x1)
Q: What is the difference between the point-slope form and the slope-intercept form of a linear equation?
A: The point-slope form and the slope-intercept form are two different ways to write a linear equation. The point-slope form is given by the equation:
y - y1 = m(x - x1)
while the slope-intercept form is given by the equation:
y = mx + b
where m is the slope and b is the y-intercept.
Q: How do I convert the point-slope form to the slope-intercept form?
A: To convert the point-slope form to the slope-intercept form, you can simplify the equation by solving for y:
y - y1 = m(x - x1) y - y1 = mx - mx1 y = mx - mx1 + y1
Q: What is the significance of the slope in a linear equation?
A: The slope in a linear equation represents the rate of change of the line. A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing.
Q: How do I find the slope of a line given two points?
A: To find the slope of a line given two points, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two points.
Q: What is the significance of the y-intercept in a linear equation?
A: The y-intercept in a linear equation represents the point where the line intersects the y-axis. It is the value of y when x is equal to 0.
Q: How do I find the y-intercept of a line given a point and a slope?
A: To find the y-intercept of a line given a point and a slope, you can use the point-slope form of a linear equation and solve for y when x is equal to 0:
y - y1 = m(x - x1) y - y1 = m(0 - x1) y - y1 = -mx1 y = -mx1 + y1
Q: What are some real-world applications of linear equations?
A: Linear equations have many real-world applications, such as:
- Physics: Linear equations can be used to describe the motion of objects under constant acceleration.
- Engineering: Linear equations can be used to design and optimize systems, such as bridges and buildings.
- Computer Science: Linear equations can be used to develop algorithms and models for machine learning and data analysis.
Q: How do I use linear equations in real-world problems?
A: To use linear equations in real-world problems, you can:
- Model real-world situations: Use linear equations to model real-world situations, such as the motion of an object or the growth of a population.
- Solve problems: Use linear equations to solve problems, such as finding the distance between two points or the cost of a product.
- Make predictions: Use linear equations to make predictions, such as predicting the future value of a stock or the growth of a population.
Q: What are some common mistakes to avoid when working with linear equations?
A: Some common mistakes to avoid when working with linear equations include:
- Not checking the domain: Make sure to check the domain of the equation to ensure that it is valid.
- Not checking the range: Make sure to check the range of the equation to ensure that it is valid.
- Not using the correct formula: Make sure to use the correct formula for the equation, such as the point-slope form or the slope-intercept form.
Q: How do I graph a linear equation?
A: To graph a linear equation, you can:
- Plot two points: Plot two points on the graph, such as the x-intercept and the y-intercept.
- Draw a line: Draw a line through the two points to create the graph.
- Label the axes: Label the x-axis and the y-axis to create a coordinate system.
Q: What are some common applications of linear equations in science and engineering?
A: Some common applications of linear equations in science and engineering include:
- Physics: Linear equations can be used to describe the motion of objects under constant acceleration.
- Engineering: Linear equations can be used to design and optimize systems, such as bridges and buildings.
- Computer Science: Linear equations can be used to develop algorithms and models for machine learning and data analysis.
Q: How do I use linear equations in computer science?
A: To use linear equations in computer science, you can:
- Develop algorithms: Use linear equations to develop algorithms for machine learning and data analysis.
- Create models: Use linear equations to create models for machine learning and data analysis.
- Make predictions: Use linear equations to make predictions, such as predicting the future value of a stock or the growth of a population.