Question

Parallel & Perpendicular Lines Quick Check
5 of 55 of 5 Items
Question
Which line is perpendicular to the line y=5x−2
and goes through the point (2,3)
?(1 point)
Responses

y+3=15(x−2)
y plus 3 equals Start Fraction 1 over 5 End Fraction left parenthesis x minus 2 right parenthesis

y−3=5(x−2)
y minus 3 equals 5 left parenthesis x minus 2 right parenthesis

y+3=−15(x+2)
y plus 3 equals negative Start Fraction 1 over 5 End Fraction left parenthesis x plus 2 right parenthesis

y−3=−15(x−2)

Answer 1

To find which line is perpendicular to the line \( y = 5x - 2 \) and goes through the point \( (2, 3) \), we first need to find the slope of the given line.

The slope of the line \( y = 5x - 2 \) is \( 5 \). The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line will be:

\[
-\frac{1}{5}
\]

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

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

Where \( m \) is the slope, and \( (x_1, y_1) \) is the point through which the line passes. In this case, \( (x_1, y_1) = (2, 3) \) and \( m = -\frac{1}{5} \).

Substituting these values into the point-slope form:

\[
y - 3 = -\frac{1}{5}(x - 2)
\]

Now we can select the corresponding response from the options given:

1. \( y + 3 = \frac{1}{5}(x - 2) \)
2. \( y - 3 = 5(x - 2) \)
3. \( y + 3 = -\frac{1}{5}(x + 2) \)
4. \( y - 3 = -\frac{1}{5}(x - 2) \)

Looking at the response options, the correct equation that matches our derived equation \( y - 3 = -\frac{1}{5}(x - 2) \) is:

**Response: \( y - 3 = -\frac{1}{5}(x - 2) \)**.