Given the line y=3x+4 and the point (2,−1) , which of the following equations represents a line perpendicular to the given line and passing through the specified point?(1 point)

Question

Responses

y=−13x−13
y is equal to negative 1 third x minus 1 third

y=3x−7
y is equal to 3 x minus 7

y=−3x+5
y is equal to negative 3 x plus 5

y=13x−73

Answer 1

To find the equation of a line that is perpendicular to the given line \(y = 3x + 4\) and passes through the point \((2, -1)\), we need to follow these steps:

1. Find the slope of the given line. The slope from the equation \(y = 3x + 4\) is \(3\).

2. The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the perpendicular slope \(m\) is given by:

\[ m = -\frac{1}{3} \]

3. Now we will use the point-slope form of the equation of a line, which is:

\[ y - y_1 = m(x - x_1) \]

where \((x_1, y_1)\) is the point \((2, -1)\). Plugging in the values:

\[ y - (-1) = -\frac{1}{3}(x - 2) \]

4. Simplifying this, we have:

\[ y + 1 = -\frac{1}{3}x + \frac{2}{3} \]

5. Now, subtracting \(1\) (which is equivalent to \(\frac{3}{3}\)) from both sides gives:

\[ y = -\frac{1}{3}x + \frac{2}{3} - \frac{3}{3} \] \[ y = -\frac{1}{3}x - \frac{1}{3} \]

6. Thus, the equation of the line we are looking for is:

\[ y = -\frac{1}{3}x - \frac{1}{3} \]

Now, looking at the provided response options, the equation that matches this formula is:

y is equal to negative one third x minus one third.