Question

the equation of line c is $y - 6 = -\frac{7}{6}(x - 5)$. line d, which is parallel to line c, includes the point (5, -7). what is the equation of line d?
write the equation in slope - intercept form. write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Explanation:

Step1: Determine slope of line d

Parallel lines have equal slopes. The equation of line \( c \) is in point - slope form \( y - y_1=m(x - x_1) \), where \( m \) is the slope. For line \( c \), \( m =-\frac{7}{6} \), so the slope of line \( d \), \( m_d=-\frac{7}{6} \).

Step2: Use point - slope form for line d

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(5, - 7) \) and \( m =-\frac{7}{6} \). Substitute these values into the formula:
\( y-(-7)=-\frac{7}{6}(x - 5) \)
Simplify the left - hand side: \( y + 7=-\frac{7}{6}(x - 5) \)

Step3: Convert to slope - intercept form (\( y=mx + b \))

Expand the right - hand side: \( y+7=-\frac{7}{6}x+\frac{35}{6} \)
Subtract 7 from both sides. Since \( 7=\frac{42}{6} \), we have:
\( y=-\frac{7}{6}x+\frac{35}{6}-\frac{42}{6} \)
Simplify the right - hand side: \( y=-\frac{7}{6}x-\frac{7}{6} \)

Answer:

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