Question

Find the equation of a line perpendicular toy, minus, 7, equals, minus, xy−7=−xthat passes through the point left parenthesis, 9, comma, 8, right parenthesis(9,8).

Answer 1

To find the equation of a line that is perpendicular to the line described by the equation \( y - 7 = -x \) and passes through the point \( (9, 8) \), we can follow these steps:

1. **Convert the given equation to slope-intercept form**:
The equation \( y - 7 = -x \) can be rearranged to find the slope:
\[
y = -x + 7
\]
Here, the slope \( m_1 \) of the line is \( -1 \).

2. **Find the slope of the perpendicular line**:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Since the slope of the original line is \( -1 \), the slope of the perpendicular line \( m_2 \) will be:
\[
m_2 = -\frac{1}{-1} = 1
\]

3. **Use the point-slope form to find the equation**:
We can use the point-slope formula \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope of the line. Plugging in the point \( (9, 8) \) and \( m = 1 \):
\[
y - 8 = 1(x - 9)
\]

4. **Simplifying the equation**:
\[
y - 8 = x - 9
\]
\[
y = x - 1
\]

Thus, the equation of the line that is perpendicular to \( y - 7 = -x \) and passes through the point \( (9, 8) \) is:
\[
\boxed{y = x - 1}
\]