Questions: Write an equation in slope-intercept form for the line parallel to y=1/2 x+5/2 containing (3/2, 1). Write an equation in slope-intercept form for the line parallel to y=8 x+11 containing (9,5). The line containing (3,5) and (-1,-3) is perpendicular to the line containing (2,5) and (m, 7). What is the value of m that satisfies this condition?

Solution Steps

1. To write an equation in slope-intercept form for the line parallel to \( y = \frac{1}{2}x + \frac{5}{2} \) containing \( \left(\frac{3}{2}, 1\right) \):

   • Identify the slope of the given line, which is \( \frac{1}{2} \).
   • Use the point-slope form of the equation \( y - y_1 = m(x - x_1) \) with the point \( \left(\frac{3}{2}, 1\right) \) and the slope \( \frac{1}{2} \).
   • Convert the equation to slope-intercept form \( y = mx + b \).

2. To write an equation in slope-intercept form for the line parallel to \( y = 8x + 11 \) containing \( (9, 5) \):

   • Identify the slope of the given line, which is \( 8 \).
   • Use the point-slope form of the equation \( y - y_1 = m(x - x_1) \) with the point \( (9, 5) \) and the slope \( 8 \).
   • Convert the equation to slope-intercept form \( y = mx + b \).

The slope of the given line is \( m_1 = \frac{1}{2} \). Using the point \( \left(\frac{3}{2}, 1\right) \), we apply the point-slope form:

\[ y - y_1 = m_1(x - x_1) \]

Substituting the values:

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

Rearranging to slope-intercept form gives:

\[ y = \frac{1}{2}x + 0.25 \]

The slope of the given line is \( m_2 = 8 \). Using the point \( (9, 5) \), we again apply the point-slope form:

\[ y - y_2 = m_2(x - x_2) \]

Substituting the values:

\[ y - 5 = 8(x - 9) \]

Rearranging to slope-intercept form gives:

\[ y = 8x - 67 \]

Final Answer