Identifying Points on the Line

Oct 13, 2025 by windailys.com 59 views

Introduction

In mathematics, a line is a set of points that extend infinitely in two directions. The equation of a line is a mathematical representation of its graph, and it can be written in various forms, such as slope-intercept form, point-slope form, or standard form. In this article, we will focus on the slope-intercept form of a line, which is given by the equation $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept.

Understanding the Equation $y=\frac{3}{4}x+5$

The given equation is in slope-intercept form, where the slope is $\frac{3}{4}$ and the y-intercept is $5$ . This means that the line passes through the point $(0, 5)$ , which is the y-intercept. The slope of the line represents the rate of change of the line, and in this case, it is $\frac{3}{4}$ . This means that for every unit increase in $x$ , the value of $y$ increases by $\frac{3}{4}$ unit.

Identifying Points on the Line

To identify a point that lies on the graph of the line, we need to substitute the x-coordinate of the point into the equation and solve for the y-coordinate. Let's consider the given options:

Option A: (5, 8)
- Substitute $x = 5$ into the equation: $y = \frac{3}{4}(5) + 5$
- Simplify the equation: $y = \frac{15}{4} + 5$
- Convert the fraction to a decimal: $y = 3.75 + 5$
- Add the numbers: $y = 8.75$
- Since the calculated y-coordinate is not equal to 8, option A is incorrect.
Option B: (1, 6)
- Substitute $x = 1$ into the equation: $y = \frac{3}{4}(1) + 5$
- Simplify the equation: $y = \frac{3}{4} + 5$
- Convert the fraction to a decimal: $y = 0.75 + 5$
- Add the numbers: $y = 5.75$
- Since the calculated y-coordinate is not equal to 6, option B is incorrect.
Option C: (-3, 3)
- Substitute $x = -3$ into the equation: $y = \frac{3}{4}(-3) + 5$
- Simplify the equation: $y = -\frac{9}{4} + 5$
- Convert the fraction to a decimal: $y = -2.25 + 5$
- Add the numbers: $y = 2.75$
- Since the calculated y-coordinate is not equal to 3, option C is incorrect.
Option D: (-4, 2)
- Substitute $x = -4$ into the equation: $y = \frac{3}{4}(-4) + 5$
- Simplify the equation: $y = -3 + 5$
- Add the numbers: $y = 2$
- Since the calculated y-coordinate is equal to 2, option D is correct.

Conclusion

In conclusion, the point that lies on the graph of the line $y=\frac{3}{4}x+5$ is (-4, 2). This is because when we substitute $x = -4$ into the equation, we get $y = 2$ , which is the y-coordinate of the point. Therefore, the correct answer is option D.

Key Takeaways

The equation of a line can be written in various forms, such as slope-intercept form, point-slope form, or standard form.
The slope-intercept form of a line is given by the equation $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept.
To identify a point that lies on the graph of a line, we need to substitute the x-coordinate of the point into the equation and solve for the y-coordinate.
The point that lies on the graph of the line $y=\frac{3}{4}x+5$ is (-4, 2).
Graph: $y=\frac{3}{4}x+5$ - Q&A =====================================

Q: What is the slope of the line $y=\frac{3}{4}x+5$ ?

A: The slope of the line $y=\frac{3}{4}x+5$ is $\frac{3}{4}$ . This means that for every unit increase in $x$ , the value of $y$ increases by $\frac{3}{4}$ unit.

Q: What is the y-intercept of the line $y=\frac{3}{4}x+5$ ?

A: The y-intercept of the line $y=\frac{3}{4}x+5$ is $5$ . This means that the line passes through the point $(0, 5)$ .

Q: How do I identify a point that lies on the graph of the line $y=\frac{3}{4}x+5$ ?

A: To identify a point that lies on the graph of the line $y=\frac{3}{4}x+5$ , you need to substitute the x-coordinate of the point into the equation and solve for the y-coordinate.

Q: What is the equation of the line in point-slope form?

A: The equation of the line in point-slope form is $y - 5 = \frac{3}{4}(x - 0)$ .

Q: What is the equation of the line in standard form?

A: The equation of the line in standard form is $4y - 3x = 20$ .

Q: How do I graph the line $y=\frac{3}{4}x+5$ ?

A: To graph the line $y=\frac{3}{4}x+5$ , you can use the slope-intercept form of the equation. Plot the y-intercept at $(0, 5)$ and use the slope to find another point on the line. Then, draw a line through the two points.

Q: What is the relationship between the slope and the y-intercept of a line?

A: The slope and the y-intercept of a line are related in that the slope represents the rate of change of the line, while the y-intercept represents the point where the line intersects the y-axis.

Q: How do I find the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you can use the point-slope form of the equation. Substitute the coordinates of one of the points into the equation and solve for the slope. Then, substitute the coordinates of the other point into the equation and solve for the y-intercept.

Q: What is the significance of the slope-intercept form of a line?

A: The slope-intercept form of a line is significant because it allows us to easily identify the slope and the y-intercept of the line. This makes it easier to graph the line and to find the equation of the line that passes through two points.

Q: How do I use the slope-intercept form of a line to solve problems?

A: To use the slope-intercept form of a line to solve problems, you can substitute the coordinates of a point into the equation and solve for the y-coordinate. This will give you the value of the y-coordinate for a given value of the x-coordinate. You can then use this information to solve problems that involve the line.

Q: What are some common applications of the slope-intercept form of a line?

A: Some common applications of the slope-intercept form of a line include:

Graphing lines
Finding the equation of a line that passes through two points
Solving problems that involve the line
Calculating the slope and the y-intercept of a line

Conclusion

In conclusion, the slope-intercept form of a line is a powerful tool that allows us to easily identify the slope and the y-intercept of a line. It is used in a variety of applications, including graphing lines, finding the equation of a line that passes through two points, and solving problems that involve the line. By understanding the slope-intercept form of a line, we can solve a wide range of problems and make informed decisions in a variety of fields.