To determine which equation represents a line that is parallel to the given line [tex]\( y = -4x - 5 \)[/tex] and passes through the point [tex]\((0,0)\)[/tex], we need to follow a few steps.
### Step 1: Identify the slope of the given line
The given line equation is [tex]\( y = -4x - 5 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Here, the slope [tex]\( m = -4 \)[/tex].
### Step 2: Find the slope of the parallel line
Lines that are parallel to each other have the same slope. Hence, the slope of the line we are looking for must also be [tex]\( -4 \)[/tex].
### Step 3: Determine the y-intercept using the given point [tex]\((0,0)\)[/tex]
We need the line to pass through the point [tex]\((0,0)\)[/tex]. We can use the slope-intercept form of the line equation [tex]\( y = mx + b \)[/tex] where [tex]\( m = -4 \)[/tex].
Substitute the point [tex]\( (0,0) \)[/tex] into the equation:
[tex]\[
0 = -4 \cdot 0 + b
\][/tex]
[tex]\[
0 = b
\][/tex]
Therefore, the y-intercept [tex]\( b \)[/tex] is 0.
### Step 4: Write the final equation
Now that we have determined the slope is [tex]\( -4 \)[/tex] and the y-intercept is 0, the equation of the line is:
[tex]\[
y = -4x
\][/tex]
### Step 5: Compare with the given options
We are to select from the following options:
1. [tex]\( y = -\frac{1}{4} x - 5 \)[/tex]
2. [tex]\( y = -4 x - 7 \)[/tex]
3. [tex]\( y = 4 x - 7 \)[/tex]
4. [tex]\( y = -4 x \)[/tex]
5. [tex]\( y = 4 x - 9 \)[/tex]
From these options, the equation [tex]\( y = -4 x \)[/tex] correctly represents the line that is parallel to [tex]\( y = -4x - 5 \)[/tex] and passes through the point [tex]\((0,0)\)[/tex].
### Conclusion
The correct choice is:
[tex]\[
\boxed{4}
\][/tex]
