To solve the problem, we need to find the slope ("pendiente", denoted as $m$) for each of the two linear equations and determine if the lines are parallel or perpendicular.

The given equations are:

1. $\ell_1: 3y - 18x - 7 = 0$
2. $\ell_2: 6y - 36x + 3 = 0$

Step 1: Rewrite both equations in slope-intercept form $y = mx + b$ to identify the slope.

For $\ell_1$:

Start with the equation: $3y - 18x - 7 = 0$ Solve for $y$: $3y = 18x + 7 \quad (\text{add } 18x + 7 \text{ on both sides})$ $y = 6x + \frac{7}{3} \quad (\text{divide by 3})$ Thus, the slope of $\ell_1$ is $m_1 = 6$.

For $\ell_2$:

Start with the equation: $6y - 36x + 3 = 0$ Solve for $y$: $6y = 36x - 3 \quad (\text{add } 36x - 3 \text{ on both sides})$ $y = 6x - \frac{1}{2} \quad (\text{divide by 6})$ Thus, the slope of $\ell_2$ is $m_2 = 6$.

Step 2: Analyze the relationship between the slopes

• If the slopes are equal ($m_1 = m_2$), the lines are parallel.
• If the product of the slopes is $-1$ ($m_1 \times m_2 = -1$), the lines are perpendicular.

Since $m_1 = m_2 = 6$, the lines $\ell_1$ and $\ell_2$ are parallel.

Let me know if you'd like further details or have any questions!

Related Questions:

1. How do you find the slope if an equation is not given in standard form?
2. What is the condition for two lines to be perpendicular?
3. Can two lines with different slopes ever be parallel?
4. What is the geometric interpretation of the slope of a line?
5. How can you determine if two lines will intersect without graphing them?

Tip:

When dealing with linear equations, rewriting them in slope-intercept form $y = mx + b$ makes it easier to directly identify the slope $m$, which is key to understanding the line's behavior.