To determine the relationship between the graphs of the given linear equations, let's first transform each equation into the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope of the line.
### Step-by-Step Solution:
Equation 1:
[tex]\[ 3x - y = 5 \][/tex]
1. Isolate [tex]\( y \)[/tex]:
[tex]\[ -y = -3x + 5 \][/tex]
2. Multiply by -1 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 5 \][/tex]
From this, we see that the slope ([tex]\( m \)[/tex]) of the first line is:
[tex]\[ m_1 = 3 \][/tex]
Equation 2:
[tex]\[ 3x + 6y = 24 \][/tex]
1. Isolate [tex]\( 6y \)[/tex]:
[tex]\[ 6y = -3x + 24 \][/tex]
2. Divide the entire equation by 6 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{2}x + 4 \][/tex]
From this, we see that the slope ([tex]\( m \)[/tex]) of the second line is:
[tex]\[ m_2 = -\frac{1}{2} \][/tex]
### Comparing the Slopes:
- The slope of the first line is [tex]\( m_1 = 3 \)[/tex].
- The slope of the second line is [tex]\( m_2 = -\frac{1}{2} \)[/tex].
To understand the relationship between the two lines, we consider the following:
1. Parallel Lines: If the slopes are equal ([tex]\( m_1 = m_2 \)[/tex]), the lines are parallel.
2. Perpendicular Lines: If the product of the slopes ([tex]\( m_1 \times m_2 \)[/tex]) equals -1, the lines are perpendicular.
### Calculating the Product of the Slopes:
[tex]\[ m_1 \times m_2 = 3 \times -\frac{1}{2} = -\frac{3}{2} \][/tex]
Since [tex]\( -\frac{3}{2} \neq -1 \)[/tex] and the slopes are not equal (3 ≠ -0.5), the lines are neither parallel nor perpendicular.
### Conclusion:
The best description for the relationship between the graphs of the given equations is:
- Intersecting but not perpendicular
Therefore, the correct answer is: intersecting but not perpendicular.
