Identifying the Point on the Line

Oct 13, 2025 by windailys.com 67 views

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Understanding the Problem

To find a point on the line and the line's slope, we need to analyze the given equation and identify the necessary information. The equation provided is in the form of $y + 2 = \frac{1}{4}(x + 5)$ . Our goal is to determine a point on the line and the slope of the line.

Identifying the Point on the Line

To find a point on the line, we need to identify the coordinates of the point. The coordinates of a point on the line can be found by substituting a value of $x$ into the equation and solving for $y$ . Let's choose a value of $x$ and substitute it into the equation.

Choosing a Value of $x$

For simplicity, let's choose $x = 0$ . Substituting $x = 0$ into the equation, we get:

y + 2 = \frac{1}{4}(0 + 5)

Simplifying the equation, we get:

y + 2 = \frac{5}{4}

Subtracting 2 from both sides, we get:

y = \frac{5}{4} - 2

Simplifying further, we get:

y = \frac{5}{4} - \frac{8}{4}

Combining the fractions, we get:

y = -\frac{3}{4}

So, the point on the line is $(0, -\frac{3}{4})$ .

Identifying the Slope of the Line

To find the slope of the line, we need to identify the coefficient of $x$ in the equation. The coefficient of $x$ is $\frac{1}{4}$ , which represents the slope of the line.

Understanding the Slope

The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run). In this case, the slope is $\frac{1}{4}$ , which means that for every 1 unit of horizontal change, the line rises by $\frac{1}{4}$ unit.

Conclusion

In conclusion, we have found a point on the line and the slope of the line. The point on the line is $(0, -\frac{3}{4})$ , and the slope of the line is $\frac{1}{4}$ . This information can be used to graph the line and understand its behavior.

Graphing the Line

To graph the line, we can use the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope. Substituting the values we found earlier, we get:

y - (-\frac{3}{4}) = \frac{1}{4}(x - 0)

Simplifying the equation, we get:

y + \frac{3}{4} = \frac{1}{4}x

Subtracting $\frac{3}{4}$ from both sides, we get:

y = \frac{1}{4}x - \frac{3}{4}

This is the equation of the line in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Graphing the Line

To graph the line, we can plot the point $(0, -\frac{3}{4})$ and use the slope to determine the direction of the line. Since the slope is $\frac{1}{4}$ , the line will rise by $\frac{1}{4}$ unit for every 1 unit of horizontal change.

Plotting the Point

To plot the point $(0, -\frac{3}{4})$ , we can start at the origin (0, 0) and move $\frac{3}{4}$ units down. This will give us the point $(0, -\frac{3}{4})$ .

Determining the Direction of the Line

To determine the direction of the line, we can use the slope to determine the direction of the line. Since the slope is $\frac{1}{4}$ , the line will rise by $\frac{1}{4}$ unit for every 1 unit of horizontal change. This means that the line will have a gentle slope and will rise slowly as we move to the right.

Conclusion

In conclusion, we have graphed the line and understood its behavior. The line has a point on it at $(0, -\frac{3}{4})$ and a slope of $\frac{1}{4}$ . This information can be used to understand the behavior of the line and to make predictions about its behavior.

Real-World Applications

The concept of finding a point on a line and the line's slope has many real-world applications. For example, in physics, the slope of a line can be used to determine the velocity of an object. In engineering, the slope of a line can be used to determine the angle of a ramp or a slope.

Example 1: Velocity of an Object

Suppose we have an object moving at a constant velocity. We can use the slope of the line to determine the velocity of the object. If the slope of the line is $\frac{1}{4}$ , then the velocity of the object is $\frac{1}{4}$ unit per unit of time.

Example 2: Angle of a Ramp

Suppose we have a ramp with a slope of $\frac{1}{4}$ . We can use the slope of the line to determine the angle of the ramp. If the slope of the line is $\frac{1}{4}$ , then the angle of the ramp is $\tan^{-1}(\frac{1}{4})$ .

Conclusion

In conclusion, we have discussed the concept of finding a point on a line and the line's slope. We have used the equation $y + 2 = \frac{1}{4}(x + 5)$ to find a point on the line and the slope of the line. We have also graphed the line and understood its behavior. The concept of finding a point on a line and the line's slope has many real-world applications, including determining the velocity of an object and the angle of a ramp.

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Q: What is the equation of the line?

A: The equation of the line is $y + 2 = \frac{1}{4}(x + 5)$ .

Q: How do I find a point on the line?

A: To find a point on the line, you need to substitute a value of $x$ into the equation and solve for $y$ . Let's choose $x = 0$ and substitute it into the equation. We get:

y + 2 = \frac{1}{4}(0 + 5)

Simplifying the equation, we get:

y + 2 = \frac{5}{4}

Subtracting 2 from both sides, we get:

y = \frac{5}{4} - 2

Simplifying further, we get:

y = \frac{5}{4} - \frac{8}{4}

Combining the fractions, we get:

y = -\frac{3}{4}

So, the point on the line is $(0, -\frac{3}{4})$ .

Q: How do I find the slope of the line?

A: To find the slope of the line, you need to identify the coefficient of $x$ in the equation. The coefficient of $x$ is $\frac{1}{4}$ , which represents the slope of the line.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: How do I graph the line?

A: To graph the line, you can use the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope. Substituting the values we found earlier, we get:

y - (-\frac{3}{4}) = \frac{1}{4}(x - 0)

Simplifying the equation, we get:

y + \frac{3}{4} = \frac{1}{4}x

Subtracting $\frac{3}{4}$ from both sides, we get:

y = \frac{1}{4}x - \frac{3}{4}

This is the equation of the line in slope-intercept form.

Q: What are some real-world applications of finding a point on the line and the line's slope?

A: The concept of finding a point on the line and the line's slope has many real-world applications, including determining the velocity of an object and the angle of a ramp.

Q: How do I determine the velocity of an object?

A: To determine the velocity of an object, you can use the slope of the line. If the slope of the line is $\frac{1}{4}$ , then the velocity of the object is $\frac{1}{4}$ unit per unit of time.

Q: How do I determine the angle of a ramp?

A: To determine the angle of a ramp, you can use the slope of the line. If the slope of the line is $\frac{1}{4}$ , then the angle of the ramp is $\tan^{-1}(\frac{1}{4})$ .

Q: What are some common mistakes to avoid when finding a point on the line and the line's slope?

A: Some common mistakes to avoid when finding a point on the line and the line's slope include:

Not substituting a value of $x$ into the equation to find a point on the line.
Not identifying the coefficient of $x$ in the equation to find the slope of the line.
Not using the point-slope form of a linear equation to graph the line.
Not considering the real-world applications of finding a point on the line and the line's slope.

Q: How can I practice finding a point on the line and the line's slope?

A: You can practice finding a point on the line and the line's slope by:

Substituting different values of $x$ into the equation to find different points on the line.
Identifying the coefficient of $x$ in the equation to find the slope of the line.
Graphing the line using the point-slope form of a linear equation.
Considering the real-world applications of finding a point on the line and the line's slope.

Q: What are some resources for learning more about finding a point on the line and the line's slope?

A: Some resources for learning more about finding a point on the line and the line's slope include:

Online tutorials and videos
Math textbooks and workbooks
Online math communities and forums
Math apps and software

Conclusion

In conclusion, finding a point on the line and the line's slope is an important concept in mathematics that has many real-world applications. By understanding the equation of the line, finding a point on the line, and identifying the slope of the line, you can graph the line and understand its behavior. Additionally, by considering the real-world applications of finding a point on the line and the line's slope, you can develop a deeper understanding of the concept and its importance in various fields.