Determine The Intercepts Of The Line.Do Not Round Your Answers.$\[ Y = -3x + 12 \\]- Y-intercept: \[$(\square, \square)\$\]- X-intercept: \[$(\square, \square)\$\]
Understanding the Problem
In this problem, we are given a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept. Our goal is to find the x-intercept and y-intercept of the line represented by this equation.
What are Intercepts?
Intercepts are the points at which a line intersects the x-axis and y-axis. The x-intercept is the point at which the line crosses the x-axis, and the y-intercept is the point at which the line crosses the y-axis.
Finding the Y-Intercept
The y-intercept is the point at which the line crosses the y-axis. Since the y-axis is vertical, the x-coordinate of the y-intercept is always 0. To find the y-intercept, we can substitute x = 0 into the equation.
y = -3x + 12
y = -3(0) + 12
y = 12
Therefore, the y-intercept is (0, 12).
Finding the X-Intercept
The x-intercept is the point at which the line crosses the x-axis. Since the x-axis is horizontal, the y-coordinate of the x-intercept is always 0. To find the x-intercept, we can substitute y = 0 into the equation and solve for x.
y = -3x + 12
0 = -3x + 12
3x = 12
x = 4
Therefore, the x-intercept is (4, 0).
Conclusion
In this problem, we found the y-intercept and x-intercept of the line represented by the equation y = -3x + 12. The y-intercept is (0, 12), and the x-intercept is (4, 0).
Key Takeaways
- The y-intercept is the point at which the line crosses the y-axis.
- The x-intercept is the point at which the line crosses the x-axis.
- To find the y-intercept, substitute x = 0 into the equation.
- To find the x-intercept, substitute y = 0 into the equation and solve for x.
Real-World Applications
Understanding intercepts is crucial in various real-world applications, such as:
- Physics: When modeling the motion of objects, intercepts can represent the point at which an object collides with a surface or reaches a certain velocity.
- Engineering: In designing structures, intercepts can represent the point at which a beam or column intersects with a support or foundation.
- Economics: In modeling economic systems, intercepts can represent the point at which a market reaches equilibrium or a certain level of production.
Practice Problems
- Find the y-intercept and x-intercept of the line represented by the equation y = 2x - 5.
- Find the y-intercept and x-intercept of the line represented by the equation y = -x + 3.
- Find the y-intercept and x-intercept of the line represented by the equation y = 3x + 2.
Solutions
- y-intercept: (0, -5) x-intercept: (2.5, 0)
- y-intercept: (0, 3) x-intercept: (-3, 0)
- y-intercept: (0, 2) x-intercept: (-2/3, 0)
Conclusion
Q: What is the difference between the y-intercept and x-intercept?
A: The y-intercept is the point at which the line crosses the y-axis, and the x-intercept is the point at which the line crosses the x-axis.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept, substitute x = 0 into the equation and solve for y.
Q: How do I find the x-intercept of a line?
A: To find the x-intercept, substitute y = 0 into the equation and solve for x.
Q: What is the significance of the y-intercept in real-world applications?
A: The y-intercept represents the starting point or initial value of a system or process. For example, in economics, the y-intercept can represent the initial level of production or consumption.
Q: What is the significance of the x-intercept in real-world applications?
A: The x-intercept represents the point at which a system or process reaches a certain level or threshold. For example, in physics, the x-intercept can represent the point at which an object reaches a certain velocity or distance.
Q: Can a line have multiple intercepts?
A: No, a line can only have one y-intercept and one x-intercept.
Q: Can a line have no intercepts?
A: Yes, a line can have no intercepts if it is a horizontal or vertical line.
Q: How do I graph a line with intercepts?
A: To graph a line with intercepts, plot the y-intercept and x-intercept on a coordinate plane and draw a line through them.
Q: What is the relationship between the slope and intercepts of a line?
A: The slope of a line is related to the intercepts by the equation y = mx + b, where m is the slope and b is the y-intercept.
Q: Can I use the intercepts to determine the slope of a line?
A: Yes, you can use the intercepts to determine the slope of a line by rearranging the equation y = mx + b to solve for m.
Q: What are some common mistakes to avoid when finding intercepts?
A: Some common mistakes to avoid when finding intercepts include:
- Not substituting the correct value for x or y
- Not solving for the correct variable
- Not checking the units of the intercepts
Q: How do I check my work when finding intercepts?
A: To check your work, substitute the intercepts back into the original equation to ensure that they satisfy the equation.
Q: What are some real-world applications of intercepts?
A: Some real-world applications of intercepts include:
- Physics: modeling the motion of objects
- Engineering: designing structures
- Economics: modeling economic systems
- Computer Science: modeling algorithms and data structures
Conclusion
In conclusion, intercepts are a fundamental concept in mathematics and have numerous real-world applications. By understanding how to find the y-intercept and x-intercept, we can model and analyze various phenomena in physics, engineering, and economics.