To determine which lines have a [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex], we need to evaluate each given line equation by substituting [tex]\( x = 0 \)[/tex] and examining the resulting [tex]\( y \)[/tex]-value.
A line in the slope-intercept form is written as [tex]\( y = mx + b \)[/tex], where [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept. Specifically, for the point [tex]\((0, 4)\)[/tex], the [tex]\( y \)[/tex]-value should equal 4 when [tex]\( x = 0 \)[/tex].
Let's analyze each option one by one:
### Option A: [tex]\( y = -3x + 4 \)[/tex]
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -3(0) + 4 = 4 \][/tex]
The [tex]\( y \)[/tex]-intercept is 4.
### Option B: [tex]\( y = 4x + 7 \)[/tex]
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4(0) + 7 = 7 \][/tex]
The [tex]\( y \)[/tex]-intercept is 7, which does not match [tex]\( y = 4 \)[/tex].
### Option C: [tex]\( y = 5 + 4x \)[/tex]
Rewrite it in slope-intercept form for clarity:
[tex]\[ y = 4x + 5 \][/tex]
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4(0) + 5 = 5 \][/tex]
The [tex]\( y \)[/tex]-intercept is 5, which does not match [tex]\( y = 4 \)[/tex].
### Option D: [tex]\( y = 4 - x \)[/tex]
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4 - 0 = 4 \][/tex]
The [tex]\( y \)[/tex]-intercept is 4.
Based on our evaluations, the lines that have a [tex]\( y \)[/tex]-intercept at [tex]\((0, 4)\)[/tex] are:
- Option A: [tex]\( y = -3x + 4 \)[/tex]
- Option D: [tex]\( y = 4 - x \)[/tex]
Therefore, the correct answers are:
1. Option A
2. Option D
