What Is The Equation Of A Line That Is Perpendicular To Y = 2 X + 4 Y = 2x + 4 Y = 2 X + 4 And Passes Through The Point ( 4 , 6 (4, 6 ( 4 , 6 ]?

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Introduction

In mathematics, the concept of lines and their equations is a fundamental aspect of geometry and algebra. When dealing with lines, it's essential to understand how to find the equation of a line that is perpendicular to a given line and passes through a specific point. In this article, we will explore the process of finding the equation of a line that is perpendicular to the line $y = 2x + 4$ and passes through the point $(4, 6)$.

Understanding the Given Line

The given line is represented by the equation $y = 2x + 4$. This is a linear equation in the slope-intercept form, where the slope is $2$ and the y-intercept is $4$. The slope of a line is a measure of how steep it is, and in this case, the slope is positive, indicating that the line slopes upward from left to right.

Finding the Slope of the Perpendicular Line

To find the equation of a line that is perpendicular to the given line, we need to find the slope of the perpendicular line. The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. In this case, the slope of the given line is $2$, so the slope of the perpendicular line is $-\frac{1}{2}$.

Using the Point-Slope Form

Now that we have the slope of the perpendicular line, we can use the point-slope form to find its equation. The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. In this case, we are given the point $(4, 6)$, so we can substitute these values into the equation.

Substituting Values into the Point-Slope Form

Substituting the values of the point $(4, 6)$ and the slope $-\frac{1}{2}$ into the point-slope form, we get:

$$y - 6 = -\frac{1}{2}(x - 4)$$

Simplifying the Equation

To simplify the equation, we can start by distributing the slope to the terms inside the parentheses:

$$y - 6 = -\frac{1}{2}x + 2$$

Adding 6 to Both Sides

Next, we can add $6$ to both sides of the equation to isolate the term with the variable:

$$y = -\frac{1}{2}x + 8$$

Conclusion

In this article, we have explored the process of finding the equation of a line that is perpendicular to the line $y = 2x + 4$ and passes through the point $(4, 6)$. We started by understanding the given line and finding the slope of the perpendicular line. We then used the point-slope form to find the equation of the perpendicular line and simplified the equation to its final form. The equation of the perpendicular line is $y = -\frac{1}{2}x + 8$.

Example Use Case

The equation of a line that is perpendicular to $y = 2x + 4$ and passes through the point $(4, 6)$ has many practical applications in real-world scenarios. For instance, in engineering, the equation of a line can be used to design and build structures such as bridges and buildings. In physics, the equation of a line can be used to model the motion of objects and predict their trajectories.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Understand the given line: The given line is represented by the equation $y = 2x + 4$.
2. Find the slope of the perpendicular line: The slope of the perpendicular line is the negative reciprocal of the slope of the given line, which is $-\frac{1}{2}$.
3. Use the point-slope form: The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
4. Substitute values into the point-slope form: Substitute the values of the point $(4, 6)$ and the slope $-\frac{1}{2}$ into the point-slope form.
5. Simplify the equation: Simplify the equation by distributing the slope to the terms inside the parentheses and adding $6$ to both sides.

Frequently Asked Questions

• What is the equation of a line that is perpendicular to $y = 2x + 4$ and passes through the point $(4, 6)$? The equation of the perpendicular line is $y = -\frac{1}{2}x + 8$.
• How do I find the slope of the perpendicular line? The slope of the perpendicular line is the negative reciprocal of the slope of the given line.
• What is the point-slope form? The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
  

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Introduction

In our previous article, we explored the process of finding the equation of a line that is perpendicular to the line $y = 2x + 4$ and passes through the point $(4, 6)$. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the equation of a line that is perpendicular to $y = 2x + 4$ and passes through the point $(4, 6)$?

A: The equation of the perpendicular line is $y = -\frac{1}{2}x + 8$.

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

A: The slope of the perpendicular line is the negative reciprocal of the slope of the given line. In this case, the slope of the given line is $2$, so the slope of the perpendicular line is $-\frac{1}{2}$.

Q: What is the point-slope form?

A: The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q: How do I use the point-slope form to find the equation of a line?

A: To use the point-slope form, you need to substitute the values of the point and the slope into the equation. In this case, we substituted the values of the point $(4, 6)$ and the slope $-\frac{1}{2}$ into the point-slope form.

Q: Can I use the point-slope form to find the equation of a line that is parallel to the given line?

A: No, the point-slope form is used to find the equation of a line that is perpendicular to the given line, not parallel.

Q: How do I find the equation of a line that is parallel to the given line?

A: To find the equation of a line that is parallel to the given line, you need to use the slope-intercept form and substitute the slope of the given line into the equation.

Q: What is the difference between the slope-intercept form and the point-slope form?

A: The slope-intercept form is given by the equation $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ 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 that is tangent to a curve?

A: No, the point-slope form is used to find the equation of a line that is perpendicular to the given line, not tangent to a curve.

Q: How do I find the equation of a line that is tangent to a curve?

A: To find the equation of a line that is tangent to a curve, you need to use the derivative of the curve and substitute the point of tangency into the equation.

Conclusion

In this article, we have answered some frequently asked questions related to finding the equation of a line that is perpendicular to the line $y = 2x + 4$ and passes through the point $(4, 6)$. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the topic.

Example Use Case

The equation of a line that is perpendicular to $y = 2x + 4$ and passes through the point $(4, 6)$ has many practical applications in real-world scenarios. For instance, in engineering, the equation of a line can be used to design and build structures such as bridges and buildings. In physics, the equation of a line can be used to model the motion of objects and predict their trajectories.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Understand the given line: The given line is represented by the equation $y = 2x + 4$.
2. Find the slope of the perpendicular line: The slope of the perpendicular line is the negative reciprocal of the slope of the given line, which is $-\frac{1}{2}$.
3. Use the point-slope form: The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.
4. Substitute values into the point-slope form: Substitute the values of the point $(4, 6)$ and the slope $-\frac{1}{2}$ into the point-slope form.
5. Simplify the equation: Simplify the equation by distributing the slope to the terms inside the parentheses and adding $6$ to both sides.

Frequently Asked Questions

• What is the equation of a line that is perpendicular to $y = 2x + 4$ and passes through the point $(4, 6)$? The equation of the perpendicular line is $y = -\frac{1}{2}x + 8$.
• How do I find the slope of the perpendicular line? The slope of the perpendicular line is the negative reciprocal of the slope of the given line.
• What is the point-slope form? The point-slope form is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.