To determine the relationship between the graphs of the equations:
[tex]\[
5y = 3x + 1
\][/tex]
[tex]\[
3y = -5x - 5
\][/tex]
we need to analyze the slopes of the lines. Here are the steps to determine the slopes and their relationship:
1. Convert each equation to slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope:
- For the first equation [tex]\( 5y = 3x + 1 \)[/tex]:
[tex]\[
y = \frac{3}{5}x + \frac{1}{5}
\][/tex]
The slope [tex]\( m_1 \)[/tex] is [tex]\( \frac{3}{5} \)[/tex].
- For the second equation [tex]\( 3y = -5x - 5 \)[/tex]:
[tex]\[
y = \frac{-5}{3}x - \frac{5}{3}
\][/tex]
The slope [tex]\( m_2 \)[/tex] is [tex]\( \frac{-5}{3} \)[/tex].
2. Determine the relationship between the slopes:
- Slopes of the lines:
[tex]\[
m_1 = \frac{3}{5}
\][/tex]
[tex]\[
m_2 = \frac{-5}{3}
\][/tex]
- Calculate the product of the slopes:
[tex]\[
m_1 \cdot m_2 = \left(\frac{3}{5}\right) \cdot \left(\frac{-5}{3}\right) = -1
\][/tex]
3. Assess the relationship based on the product of the slopes:
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
- Two lines are parallel if their slopes are equal.
- If neither condition is met, the lines are neither.
Since the product of these slopes is [tex]\(-1\)[/tex], we can conclude that the lines are perpendicular.
The correct answer is:
\textbf{perpendicular}
