To determine if the two lines are parallel, perpendicular, or neither, we need to find the slope of each line by converting both equations into slope-intercept form $y = mx + b$, where $m$ represents the slope.

Step 1: Simplify the first equation

The first equation is given as:

$\frac{6y + 5x}{2} = x + 1$

Multiply both sides by 2 to eliminate the denominator:

$6y + 5x = 2(x + 1)$

Expand the right-hand side:

$6y + 5x = 2x + 2$

Now, isolate the $y$-term by moving $5x$ and 2x to the other side:

$6y = 2x - 5x + 2$

Simplify:

$6y = -3x + 2$

Finally, divide by 6 to solve for $y$:

$y = \frac{-3}{6}x + \frac{2}{6}$

$y = -\frac{1}{2}x + \frac{1}{3}$

So, the slope of the first line is $m_1 = -\frac{1}{2}$.

Step 2: Simplify the second equation

The second equation is:

$2x - 3y = 8x + 10$

Move all $x$-terms to one side:

$-3y = 8x - 2x + 10$

Simplify:

$-3y = 6x + 10$

Now, divide by -3 to solve for $y$:

$y = \frac{-6}{-3}x + \frac{-10}{-3}$

$y = 2x - \frac{10}{3}$

So, the slope of the second line is $m_2 = 2$.

Step 3: Compare the slopes

• The slope of the first line is $m_1 = -\frac{1}{2}$.
• The slope of the second line is $m_2 = 2$.

To check the relationship between the lines:

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are negative reciprocals, meaning $m_1 \times m_2 = -1$.

In this case:

$m_1 \times m_2 = -\frac{1}{2} \times 2 = -1$

Since the product of the slopes is -1, the lines are perpendicular.

Do you have any questions or need further clarification?

Related Questions:

1. How do you convert a general linear equation into slope-intercept form?
2. What is the significance of the slope when determining the relationship between two lines?
3. Can two lines with different slopes ever be parallel?
4. How can you algebraically prove that two lines are perpendicular using their slopes?
5. In what situations would neither "parallel" nor "perpendicular" apply?

Tip:

When working with linear equations, always check that the equation is simplified correctly before interpreting the slope!