https://r2.erweima.ai/mathbot/image/1719409999958o6yxtxnf.jpg

To find the value of \( k \) for which the lines \( 4x + 2ky + 5 = 0 \) and \( 12x + 6y + 15 = 0 \) are parallel, we need to compare their slopes.

First, rewrite both equations in slope-intercept form \( y = mx + b \) to find the slopes.

1. For the line \( 4x + 2ky + 5 = 0 \):
\[ 2ky = -4x - 5 \]
\[ y = \frac{-4}{2k}x - \frac{5}{2k} \]
\[ y = \frac{-2}{k}x - \frac{5}{2k} \]
The slope \( m_1 = \frac{-2}{k} \).

2. For the line \( 12x + 6y + 15 = 0 \):
\[ 6y = -12x - 15 \]
\[ y = \frac{-12}{6}x - \frac{15}{6} \]
\[ y = -2x - \frac{5}{2} \]
The slope \( m_2 = -2 \).

Since the lines are parallel, their slopes must be equal. Therefore:
\[ \frac{-2}{k} = -2 \]

Solve for \( k \):
\[ \frac{-2}{k} = -2 \]
\[ \frac{2}{k} = 2 \]
\[ 2 = 2k \]
\[ k = 1 \]

Thus, the value of \( k \) is \( 1 \).

Would you like any further details or have any other questions?

Here are some questions you might find useful:
1. How do you find the slope of a line given its equation?
2. What is the general form of the equation of a line?
3. How can you determine if two lines are parallel or perpendicular?
4. What is the significance of the slope-intercept form of a line?
5. How do you convert from the general form to the slope-intercept form of a line?

**Tip:** When comparing the slopes of lines to check for parallelism or perpendicularity, always ensure the equations are simplified to their slope-intercept form.

Learn how to find the value of k for which the lines 4x + 2ky + 5 = 0 and 12x + 6y + 15 = 0 are parallel with a step-by-step solution. This problem involves determining slopes of lines and applying the condition for parallelism. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation