In The \[$ Xy \$\]-plane, The Slope Of The Line \[$ Y = Mx - 4 \$\] Is Less Than The Slope Of The Line \[$ Y = X - 4 \$\]. Which Of The Following Must Be True About \[$ M \$\]?A. \[$ M = -1 \$\] B. \[$ M = 1
In the xy-plane, the Slope of the Line y = mx - 4 is Less than the Slope of the Line y = x - 4
Understanding the Problem
The problem presents two lines in the xy-plane, given by the equations y = mx - 4 and y = x - 4. We are asked to determine which of the following must be true about the slope m of the line y = mx - 4, given that its slope is less than the slope of the line y = x - 4.
Recalling the Concept of Slope
The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In the case of the line y = mx - 4, the slope is represented by the coefficient m. Similarly, the slope of the line y = x - 4 is 1, since the coefficient of x is 1.
Comparing the Slopes
To compare the slopes of the two lines, we can rewrite the equation of the line y = x - 4 in the form y = mx - 4. By doing so, we get y = x - 4 = (1)x - 4. This shows that the slope of the line y = x - 4 is 1.
Analyzing the Relationship Between the Slopes
Since the slope of the line y = mx - 4 is less than the slope of the line y = x - 4, we can write the inequality m < 1. This means that the slope m of the line y = mx - 4 must be less than 1.
Evaluating the Answer Choices
Now, let's evaluate the answer choices to determine which one must be true about the slope m.
A. m = -1: This is not necessarily true, since m can be any value less than 1, not just -1.
B. m = 1: This is not true, since m must be less than 1.
C. m < 1: This is the correct answer, since the slope m of the line y = mx - 4 must be less than 1.
Conclusion
In conclusion, the slope m of the line y = mx - 4 must be less than 1, given that its slope is less than the slope of the line y = x - 4. This is the only answer choice that must be true about the slope m.
Key Takeaways
- The slope of a line is a measure of how steep it is.
- The slope of the line y = x - 4 is 1.
- The slope m of the line y = mx - 4 must be less than 1.
- The correct answer choice is C. m < 1.
Additional Examples
- Consider the line y = 2x - 4. What is its slope?
- Consider the line y = -3x - 4. What is its slope?
- Consider the line y = 0.5x - 4. What is its slope?
Solving the Problem
To solve the problem, we can use the following steps:
- Recall the concept of slope and how it is calculated.
- Compare the slopes of the two lines.
- Analyze the relationship between the slopes.
- Evaluate the answer choices.
Tips and Tricks
- When comparing the slopes of two lines, make sure to rewrite the equation of one line in the form y = mx - 4.
- Use the inequality m < 1 to determine the correct answer choice.
- Make sure to evaluate all answer choices before selecting the correct one.
Common Mistakes
- Failing to rewrite the equation of one line in the form y = mx - 4.
- Not using the inequality m < 1 to determine the correct answer choice.
- Not evaluating all answer choices before selecting the correct one.
Real-World Applications
- The concept of slope is used in many real-world applications, such as architecture, engineering, and physics.
- The slope of a line can be used to determine the steepness of a roof, the angle of a ramp, or the trajectory of a projectile.
Conclusion
In conclusion, the slope m of the line y = mx - 4 must be less than 1, given that its slope is less than the slope of the line y = x - 4. This is the only answer choice that must be true about the slope m.
Q&A: In the xy-plane, the Slope of the Line y = mx - 4 is Less than the Slope of the Line y = x - 4
Frequently Asked Questions
Q: What is the slope of the line y = x - 4?
A: The slope of the line y = x - 4 is 1, since the coefficient of x is 1.
Q: How do I compare the slopes of two lines?
A: To compare the slopes of two lines, you can rewrite the equation of one line in the form y = mx - 4. Then, you can compare the coefficients of x to determine the slope.
Q: What is the relationship between the slopes of the two lines?
A: Since the slope of the line y = mx - 4 is less than the slope of the line y = x - 4, we can write the inequality m < 1.
Q: What is the correct answer choice?
A: The correct answer choice is C. m < 1, since the slope m of the line y = mx - 4 must be less than 1.
Q: Can the slope m be equal to 1?
A: No, the slope m cannot be equal to 1, since it must be less than 1.
Q: Can the slope m be greater than 1?
A: No, the slope m cannot be greater than 1, since it must be less than 1.
Q: What is the significance of the slope in real-world applications?
A: The slope of a line is used in many real-world applications, such as architecture, engineering, and physics. It can be used to determine the steepness of a roof, the angle of a ramp, or the trajectory of a projectile.
Q: How do I determine the correct answer choice?
A: To determine the correct answer choice, you can use the inequality m < 1 and evaluate all answer choices before selecting the correct one.
Q: What are some common mistakes to avoid?
A: Some common mistakes to avoid include failing to rewrite the equation of one line in the form y = mx - 4, not using the inequality m < 1 to determine the correct answer choice, and not evaluating all answer choices before selecting the correct one.
Q: Can I use the concept of slope to solve other problems?
A: Yes, you can use the concept of slope to solve other problems. For example, you can use it to determine the steepness of a roof, the angle of a ramp, or the trajectory of a projectile.
Q: How do I apply the concept of slope in real-world situations?
A: To apply the concept of slope in real-world situations, you can use it to determine the steepness of a roof, the angle of a ramp, or the trajectory of a projectile. You can also use it to design buildings, bridges, or other structures.
Q: What are some real-world applications of the concept of slope?
A: Some real-world applications of the concept of slope include architecture, engineering, and physics. It can be used to determine the steepness of a roof, the angle of a ramp, or the trajectory of a projectile.
Q: Can I use the concept of slope to solve problems in other subjects?
A: Yes, you can use the concept of slope to solve problems in other subjects, such as physics, engineering, and architecture.
Q: How do I extend my knowledge of the concept of slope?
A: To extend your knowledge of the concept of slope, you can read more about it, practice solving problems, and apply it to real-world situations.
Q: What are some advanced topics related to the concept of slope?
A: Some advanced topics related to the concept of slope include calculus, differential equations, and linear algebra.
Q: Can I use the concept of slope to solve problems in other areas of mathematics?
A: Yes, you can use the concept of slope to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.
Conclusion
In conclusion, the concept of slope is a fundamental concept in mathematics that has many real-world applications. It can be used to determine the steepness of a roof, the angle of a ramp, or the trajectory of a projectile. By understanding the concept of slope, you can solve problems in many areas of mathematics and apply it to real-world situations.