The Equation Of Line $L_1$ Is $y = 2x - 5$. The Equation Of Line $L_2$ Is $6y + Kx - 12 = 0$. Line $L_1$ Is Perpendicular To Line $L_2$. Find The Value Of $k$. You Must Show All Your

Introduction

In this article, we will explore the concept of perpendicular lines and how to find the value of a variable in the equation of a line. We will use the given equations of two lines, $L_1$ and $L_2$, and apply the concept of perpendicularity to find the value of $k$.

The Equation of Line $L_1$

The equation of line $L_1$ is given as $y = 2x - 5$. This is a linear equation in the slope-intercept form, where the slope of the line is $2$ and the y-intercept is $-5$.

The Equation of Line $L_2$

The equation of line $L_2$ is given as $6y + kx - 12 = 0$. This is a linear equation in the general form, where the coefficients of $x$ and $y$ are $k$ and $6$, respectively.

Perpendicular Lines

Two lines are said to be perpendicular if the product of their slopes is $-1$. In other words, if the slope of one line is $m$, then the slope of the other line is $-\frac{1}{m}$.

Finding the Slope of Line $L_2$

To find the slope of line $L_2$, we need to rewrite the equation in the slope-intercept form. We can do this by isolating $y$ on one side of the equation.

# Import necessary modules
import sympy as sp

# Define variables
x, y, k = sp.symbols('x y k')

# Define the equation of line L2
eq = 6*y + k*x - 12

# Solve for y
y_expr = sp.solve(eq, y)[0]

# Print the expression for y
print(y_expr)

The output of the above code is:

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

From this expression, we can see that the slope of line $L_2$ is $-\frac{k}{6}$.

Finding the Value of $k$

Since line $L_1$ is perpendicular to line $L_2$, the product of their slopes is $-1$. We can set up an equation using this information:

$$2 \times \left(-\frac{k}{6}\right) = -1$$

Simplifying this equation, we get:

$$-\frac{k}{3} = -1$$

Multiplying both sides by $-3$, we get:

$$k = 3$$

Therefore, the value of $k$ is $3$.

Conclusion

In this article, we used the concept of perpendicular lines to find the value of $k$ in the equation of line $L_2$. We first found the slope of line $L_2$ by rewriting the equation in the slope-intercept form. Then, we used the fact that the product of the slopes of two perpendicular lines is $-1$ to set up an equation and solve for $k$. The value of $k$ is $3$.

References

• [1] Khan Academy. (n.d.). Perpendicular Lines. Retrieved from https://www.khanacademy.org/math/geometry/geometry-lines/geometry-perpendicular-lines/v/perpendicular-lines
• [2] Math Open Reference. (n.d.). Perpendicular Lines. Retrieved from https://www.mathopenref.com/lineperpendicular.html

Glossary

• Perpendicular lines: Two lines that intersect at a right angle.
• Slope: A measure of how steep a line is.
• Slope-intercept form: A way of writing a linear equation in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• General form: A way of writing a linear equation in the form $ax + by + c = 0$, where $a$, $b$, and $c$ are constants.
  The Equation of Line $L_1$ and $L_2$: Finding the Value of $k$ - Q&A ====================================================================

Introduction

In our previous article, we explored the concept of perpendicular lines and how to find the value of a variable in the equation of a line. We used the given equations of two lines, $L_1$ and $L_2$, and applied the concept of perpendicularity to find the value of $k$. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the difference between the slope-intercept form and the general 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. The general form of a linear equation is $ax + by + c = 0$, where $a$, $b$, and $c$ are constants. The slope-intercept form is useful for finding the slope and y-intercept of a line, while the general form is useful for finding the equation of a line given its slope and y-intercept.

Q: How do you find the slope of a line given its equation in the general form?

A: To find the slope of a line given its equation in the general form, you need to rewrite the equation in the slope-intercept form. This can be done by isolating $y$ on one side of the equation. For example, if the equation is $6y + kx - 12 = 0$, you can rewrite it as $y = -\frac{k}{6}x + 2$.

Q: What is the relationship between the slopes of two perpendicular lines?

A: The slopes of two perpendicular lines are negative reciprocals of each other. In other words, if the slope of one line is $m$, then the slope of the other line is $-\frac{1}{m}$.

Q: How do you find the value of $k$ in the equation of a line given its slope and the slope of a perpendicular line?

A: To find the value of $k$ in the equation of a line given its slope and the slope of a perpendicular line, you can use the fact that the product of the slopes of two perpendicular lines is $-1$. For example, if the slope of one line is $2$ and the slope of a perpendicular line is $-\frac{k}{6}$, you can set up an equation using the fact that the product of the slopes is $-1$.

Q: What is the significance of the value of $k$ in the equation of a line?

A: The value of $k$ in the equation of a line is a constant that determines the slope of the line. In the equation $6y + kx - 12 = 0$, the value of $k$ determines the slope of the line. If $k$ is positive, the line has a positive slope. If $k$ is negative, the line has a negative slope.

Q: How do you determine the equation of a line given its slope and a point on the line?

A: To determine the equation of a line given its slope and a point on the line, you can use the point-slope form of a linear equation. The point-slope form is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Conclusion

In this article, we answered some frequently asked questions related to the topic of finding the value of $k$ in the equation of a line. We hope that this article has been helpful in clarifying any doubts that you may have had.

References

• [1] Khan Academy. (n.d.). Perpendicular Lines. Retrieved from https://www.khanacademy.org/math/geometry/geometry-lines/geometry-perpendicular-lines/v/perpendicular-lines
• [2] Math Open Reference. (n.d.). Perpendicular Lines. Retrieved from https://www.mathopenref.com/lineperpendicular.html

Glossary

• Perpendicular lines: Two lines that intersect at a right angle.
• Slope: A measure of how steep a line is.
• Slope-intercept form: A way of writing a linear equation in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• General form: A way of writing a linear equation in the form $ax + by + c = 0$, where $a$, $b$, and $c$ are constants.
• Point-slope form: A way of writing a linear equation in the form $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.