What Is The { Y $}$-intercept Of The Line Given By The Equation { Y = 3x - 11 $}$?A. { (0, -11)$}$ B. { (0, -3)$}$ C. { (0, 3)$}$ D. { (0, 11)$}$
The y-intercept of a line is a crucial concept in mathematics, particularly in algebra and geometry. It represents the point at which the line intersects the y-axis, and it is denoted by the coordinates (0, y). In this article, we will explore the concept of the y-intercept and how to find it using the given equation of a line.
What is the y-intercept?
The y-intercept is the point on the y-axis where the line crosses it. It is the value of y when x is equal to 0. In other words, it is the value of y that the line reaches when it intersects the y-axis. The y-intercept is an important concept in mathematics because it helps us understand the behavior of a line and its relationship with the coordinate plane.
Finding the y-intercept
To find the y-intercept of a line, we need to use the equation of the line. The equation of a line is typically written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. In this case, the equation of the line is y = 3x - 11.
Breaking Down the Equation
Let's break down the equation y = 3x - 11 to understand its components. The equation consists of two parts: the slope (3x) and the y-intercept (-11). The slope represents the rate of change of the line, while the y-intercept represents the point at which the line intersects the y-axis.
Identifying the y-intercept
Now that we have broken down the equation, let's identify the y-intercept. The y-intercept is the value of y when x is equal to 0. In this case, when x is equal to 0, the equation becomes y = 3(0) - 11. Simplifying this expression, we get y = -11.
Conclusion
In conclusion, the y-intercept of the line given by the equation y = 3x - 11 is (0, -11). This means that the line intersects the y-axis at the point (0, -11). The y-intercept is an important concept in mathematics, and it helps us understand the behavior of a line and its relationship with the coordinate plane.
Answer
The correct answer is A. (0, -11).
Additional Examples
Here are a few additional examples to help you understand the concept of the y-intercept:
- Find the y-intercept of the line given by the equation y = 2x + 5.
- Find the y-intercept of the line given by the equation y = -4x + 2.
- Find the y-intercept of the line given by the equation y = x - 3.
Solving the Examples
Let's solve the examples:
- Find the y-intercept of the line given by the equation y = 2x + 5.
To find the y-intercept, we need to set x equal to 0 and solve for y.
y = 2(0) + 5 y = 5
The y-intercept of the line is (0, 5).
- Find the y-intercept of the line given by the equation y = -4x + 2.
To find the y-intercept, we need to set x equal to 0 and solve for y.
y = -4(0) + 2 y = 2
The y-intercept of the line is (0, 2).
- Find the y-intercept of the line given by the equation y = x - 3.
To find the y-intercept, we need to set x equal to 0 and solve for y.
y = (0) - 3 y = -3
The y-intercept of the line is (0, -3).
Conclusion
The y-intercept is a fundamental concept in mathematics, and it can be a bit confusing at first. In this article, we will answer some frequently asked questions about the y-intercept to help you better understand this concept.
Q: What is the y-intercept?
A: The y-intercept is the point on the y-axis where the line crosses it. It is the value of y when x is equal to 0.
Q: How do I find the y-intercept?
A: To find the y-intercept, you need to use the equation of the line and set x equal to 0. Then, solve for y.
Q: What is the equation of a line?
A: The equation of a line is typically written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Q: What is the slope of a line?
A: The slope of a line is the rate of change of the line. It is denoted by the letter m in the equation y = mx + b.
Q: How do I determine the y-intercept from the equation of a line?
A: To determine the y-intercept from the equation of a line, you need to look at the constant term in the equation. The constant term is the y-intercept.
Q: What is the difference between the y-intercept and the x-intercept?
A: The y-intercept is the point on the y-axis where the line crosses it, while the x-intercept is the point on the x-axis where the line crosses it.
Q: Can a line have more than one y-intercept?
A: No, a line can only have one y-intercept.
Q: Can a line have no y-intercept?
A: Yes, a line can have no y-intercept if it is a vertical line.
Q: How do I graph a line with a given y-intercept?
A: To graph a line with a given y-intercept, you need to use the y-intercept as the starting point and then draw the line using the slope.
Q: What is the significance of the y-intercept in real-life applications?
A: The y-intercept is significant in real-life applications because it represents the initial value or the starting point of a system or a process.
Q: Can you provide examples of real-life applications of the y-intercept?
A: Yes, here are a few examples of real-life applications of the y-intercept:
- In economics, the y-intercept represents the initial value of a product or a service.
- In physics, the y-intercept represents the initial velocity of an object.
- In engineering, the y-intercept represents the initial value of a system or a process.
Conclusion
In conclusion, the y-intercept is an important concept in mathematics that represents the point at which a line intersects the y-axis. It is denoted by the coordinates (0, y), where y is the value of the y-intercept. We hope that this article has helped you better understand the concept of the y-intercept and its significance in real-life applications.