Question

write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (6,9) and perpendicular to x = -7.

Explanation:

Step1: Analyze the given line

The line \( x = -7 \) is a vertical line (since it has an undefined slope, all points on this line have an x - coordinate of -7). A line perpendicular to a vertical line is a horizontal line.

Step2: Determine the equation of the horizontal line

A horizontal line has a slope of 0, and its equation is of the form \( y = k \), where \( k \) is the y - coordinate of any point on the line. The line passes through the point \( (6,9) \), so the y - coordinate \( k = 9 \).

Part (a): Slope - intercept form

The slope - intercept form of a line is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For a horizontal line with slope \( m = 0 \) and passing through \( (6,9) \), when \( x = 6 \), \( y=9 \). Substituting into \( y=mx + b \):
\( 9=0\times6 + b \), so \( b = 9 \). The equation in slope - intercept form is \( y=0x + 9 \) or simply \( y = 9 \).

Part (b): Standard form

The standard form of a line is \( Ax+By = C \), where \( A\), \( B \), and \( C \) are integers and \( A\geq0 \). For the line \( y = 9 \), we can rewrite it as \( 0x+1y=9 \).

Answer:

(a) The equation in slope - intercept form is \( \boldsymbol{y = 9} \)
(b) The equation in standard form is \( \boldsymbol{0x + y=9} \) (or simply \( \boldsymbol{y = 9} \) since the standard form \( Ax + By=C \) with \( A = 0\), \( B = 1\), \( C = 9\) is also acceptable as \( 0x+y - 9=0\) can be written as \( 0x + y=9\))