To solve for the slopes of the given pair of equations and determine the relationship between the lines, let's follow these steps:
### Step 1: Identify the equations
We have the following pair of linear equations:
[tex]\[
\begin{array}{l}
5x + 4y = 1 \\
20x + 16y = 3
\end{array}
\][/tex]
### Step 2: Put each equation in slope-intercept form ( [tex]\( y = mx + b \)[/tex] )
#### For the first equation [tex]\( 5x + 4y = 1 \)[/tex]:
1. Solve for [tex]\( y \)[/tex]:
[tex]\[
4y = -5x + 1
\][/tex]
2. Divide everything by 4:
[tex]\[
y = -\frac{5}{4}x + \frac{1}{4}
\][/tex]
Therefore, the slope ([tex]\(m\)[/tex]) of the first equation is:
[tex]\[
m_1 = -\frac{5}{4}
\][/tex]
#### For the second equation [tex]\( 20x + 16y = 3 \)[/tex]:
1. Solve for [tex]\( y \)[/tex]:
[tex]\[
16y = -20x + 3
\][/tex]
2. Divide everything by 16:
[tex]\[
y = -\frac{5}{4}x + \frac{3}{16}
\][/tex]
Therefore, the slope ([tex]\(m\)[/tex]) of the second equation is:
[tex]\[
m_2 = -\frac{5}{4}
\][/tex]
### Step 3: Determine the relationship between the lines
To check whether the lines are parallel, perpendicular, or neither:
- Lines are parallel if their slopes are equal.
- Lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
- If neither condition is met, then the lines are neither parallel nor perpendicular.
Comparing the slopes:
[tex]\[
m_1 = -\frac{5}{4}, \quad m_2 = -\frac{5}{4}
\][/tex]
Since [tex]\( m_1 = m_2 \)[/tex], the lines are parallel.
### Summary
- The slope of the first equation [tex]\( 5x + 4y = 1 \)[/tex] is [tex]\(-\frac{5}{4}\)[/tex].
- The slope of the second equation [tex]\( 20x + 16y = 3 \)[/tex] is [tex]\(-\frac{5}{4}\)[/tex].
- The lines are parallel.
So, the correct answers are:
- The slope of [tex]\( 5x + 4y = 1 \)[/tex]: [tex]\(-\frac{5}{4}\)[/tex]
- The slope of [tex]\( 20x + 16y = 3 \)[/tex]: [tex]\(-\frac{5}{4}\)[/tex]
- The lines are parallel.
