The equation for line v can be written as y=


9
7
x–5. Line w, which is perpendicular to line v, includes the point (9,6). What is the equation of line w?
Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

1 answer

To find the equation of line \( w \) that is perpendicular to line \( v \) and passes through the point \( (9, 6) \), we first need to determine the slope of line \( v \) from its equation.

The equation of line \( v \) is given as:

\[ y = -\frac{9}{7}x - 5 \]

From this, we can see that the slope \( m_v \) of line \( v \) is \( -\frac{9}{7} \).

Since line \( w \) is perpendicular to line \( v \), its slope \( m_w \) will be the negative reciprocal of \( m_v \):

\[ m_w = -\frac{1}{m_v} = -\frac{1}{-\frac{9}{7}} = \frac{7}{9} \]

Now we have the slope of line \( w \) as \( \frac{7}{9} \). Next, we can use the point-slope form of a line's equation to find the equation of line \( w \) that passes through the point \( (9, 6) \):

The point-slope form is given by:

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

Where \( (x_1, y_1) \) is a point on the line. Plugging in our point \( (9, 6) \) and the slope \( \frac{7}{9} \):

\[ y - 6 = \frac{7}{9}(x - 9) \]

Now, we will simplify this equation to get it into slope-intercept form \( y = mx + b \).

Expanding the right-hand side:

\[ y - 6 = \frac{7}{9}x - \frac{7}{9} \times 9 \] \[ y - 6 = \frac{7}{9}x - 7 \]

Now, add 6 to both sides:

\[ y = \frac{7}{9}x - 7 + 6 \] \[ y = \frac{7}{9}x - 1 \]

Thus, the equation of line \( w \) in slope-intercept form is:

\[ \boxed{y = \frac{7}{9}x - 1} \]