Nolan Plots The $y$-intercept Of A Line At $(0,3)$ On The $y$-axis. He Uses A Slope Of 2 To Graph Another Point. He Draws A Line Through The Two Points. Which Equation Represents Nolan's Line?A. $y = 2x +
Introduction
In mathematics, the equation of a line is a fundamental concept that represents the relationship between two variables, typically x and y. Given two points on a line, we can determine the equation of the line using various methods, including the slope-intercept form and the point-slope form. In this article, we will explore how to find the equation of a line when given two points, with a focus on solving for the y-intercept.
Understanding the Problem
Nolan plots the y-intercept of a line at (0,3) on the y-axis. He then uses a slope of 2 to graph another point. By drawing a line through the two points, Nolan creates a linear equation that represents the relationship between x and y. Our goal is to determine the equation of Nolan's line.
The Slope-Intercept Form
The slope-intercept form of a linear equation is given by:
y = mx + b
where m is the slope of the line, and b is the y-intercept. In this case, we are given the y-intercept as (0,3), which means that b = 3. We are also given the slope as 2, which means that m = 2.
Finding the Equation of the Line
To find the equation of the line, we can substitute the values of m and b into the slope-intercept form:
y = 2x + 3
This equation represents the relationship between x and y for Nolan's line.
The Point-Slope Form
The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope. In this case, we can use the point (0,3) as (x1, y1), and the slope as 2.
Substituting Values into the Point-Slope Form
Substituting the values into the point-slope form, we get:
y - 3 = 2(x - 0)
Simplifying the equation, we get:
y - 3 = 2x
Adding 3 to both sides of the equation, we get:
y = 2x + 3
This equation represents the relationship between x and y for Nolan's line.
Conclusion
In this article, we have explored how to find the equation of a line when given two points. We used the slope-intercept form and the point-slope form to determine the equation of Nolan's line. By substituting the values of m and b into the slope-intercept form, we obtained the equation y = 2x + 3. This equation represents the relationship between x and y for Nolan's line.
A. y = 2x + 3
This is the correct equation of Nolan's line.
B. y = 2x - 3
This is not the correct equation of Nolan's line.
C. y = 2x + 2
This is not the correct equation of Nolan's line.
D. y = 2x - 2
This is not the correct equation of Nolan's line.
Answer
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is given by:
y = mx + b
where m is the slope of the line, and b is the y-intercept.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope.
Q: How do I find the equation of a line when given two points?
A: To find the equation of a line when given two points, you can use either the slope-intercept form or the point-slope form. If you know the slope and one of the points, you can use the point-slope form. If you know the y-intercept and one of the points, you can use the slope-intercept form.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is represented by the value of b in the slope-intercept form.
Q: How do I find the slope of a line when given two points?
A: To find the slope of a line when 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 equation of a line when the slope is 2 and the y-intercept is 3?
A: The equation of a line when the slope is 2 and the y-intercept is 3 is given by:
y = 2x + 3
Q: How do I graph a line when given its equation?
A: To graph a line when given its equation, you can use the slope-intercept form or the point-slope form. If you know the y-intercept, you can graph the line by plotting the point (0, b) and then using the slope to find other points on the line.
Q: What is the difference between the slope-intercept form and the point-slope form?
A: The slope-intercept form and the point-slope form are two different ways of representing a linear equation. The slope-intercept form is given by y = mx + b, where m is the slope and b is the y-intercept. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: Can I use the point-slope form to find the equation of a line when given the y-intercept?
A: No, you cannot use the point-slope form to find the equation of a line when given the y-intercept. The point-slope form requires a point on the line and the slope, but the y-intercept is not a point on the line.
Q: Can I use the slope-intercept form to find the equation of a line when given two points?
A: Yes, you can use the slope-intercept form to find the equation of a line when given two points. You can first find the slope using the formula m = (y2 - y1) / (x2 - x1), and then use the slope-intercept form to find the equation of the line.
Q: What is the equation of a line when the slope is 2 and the y-intercept is -3?
A: The equation of a line when the slope is 2 and the y-intercept is -3 is given by:
y = 2x - 3
Q: Can I use the point-slope form to find the equation of a line when given the slope and the y-intercept?
A: No, you cannot use the point-slope form to find the equation of a line when given the slope and the y-intercept. The point-slope form requires a point on the line and the slope, but the y-intercept is not a point on the line.
Q: Can I use the slope-intercept form to find the equation of a line when given the slope and a point on the line?
A: Yes, you can use the slope-intercept form to find the equation of a line when given the slope and a point on the line. You can first use the point-slope form to find the equation of the line, and then simplify the equation to the slope-intercept form.