To determine which of the given lines is perpendicular to a certain line and passes through a specific point, we need to follow these steps:
### Step 1: Identify the slope of the given line
The given line equations are:
[tex]\[ y = -\frac{1}{3}x + 5 \][/tex]
[tex]\[ y = -\frac{1}{3}x + 3 \][/tex]
[tex]\[ y = 3x + 2 \][/tex]
[tex]\[ y = 3x - 5 \][/tex]
We need to find a line that is perpendicular to one of these lines. The general form of the equation of a line is [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
### Step 2: Identify the perpendicular slope
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
- For the slopes [tex]\(-\frac{1}{3}\)[/tex] of the first two lines:
[tex]\[ \text{Slope of the perpendicular line} = -\frac{1}{\left(-\frac{1}{3}\right)} = 3 \][/tex]
- For the slopes [tex]\(3\)[/tex] of the other two lines:
[tex]\[ \text{Slope of the perpendicular line} = -\frac{1}{3} \][/tex]
### Step 3: Check which line matches the perpendicular slope and passes through the point [tex]\((3, 4)\)[/tex]
We need to verify the lines that have a slope of 3 and see if they pass through the point [tex]\((3, 4)\)[/tex]:
The given lines with a slope of [tex]\(3\)[/tex] are:
[tex]\[ y = 3x + 2 \][/tex]
[tex]\[ y = 3x - 5 \][/tex]
Now, we'll substitute [tex]\(x = 3\)[/tex] and [tex]\(y = 4\)[/tex] into these equations to see if they satisfy the line equations.
1. Check [tex]\(y = 3x + 2\)[/tex]:
[tex]\[ 4 = 3(3) + 2 \][/tex]
[tex]\[ 4 = 9 + 2 \][/tex]
[tex]\[ 4 \neq 11 \][/tex]
So, the line [tex]\(y = 3x + 2\)[/tex] does not pass through the point [tex]\((3, 4)\)[/tex].
2. Check [tex]\(y = 3x - 5\)[/tex]:
[tex]\[ 4 = 3(3) - 5 \][/tex]
[tex]\[ 4 = 9 - 5 \][/tex]
[tex]\[ 4 = 4 \][/tex]
So, the line [tex]\(y = 3x - 5\)[/tex] does pass through the point [tex]\((3, 4)\)[/tex].
### Conclusion
The equation of the line that is perpendicular to the line [tex]\(y = -\frac{1}{3}x + 5\)[/tex] (or [tex]\(y = -\frac{1}{3}x + 3\)[/tex]) and passes through the point [tex]\((3, 4)\)[/tex] is:
[tex]\[ y = 3x - 5 \][/tex]
