Question

A line passes through the point(1,6) and is parallel to y=4x+6. what is the value of b?

Answer 1

keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above

`y= \stackrel{\stackrel{m}{\downarrow }}{4}x+6\qquad \impliedby \begin{array} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}`

so we're really looking for the equation of a line whose slope is 4 and passes through (1 , 6)

`(\stackrel{x_1}{1}~,~\stackrel{y_1}{6})\hspace{10em} \stackrel{slope}{m} ~=~ 4 \\\\\\ \begin{array}ll \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{6}=\stackrel{m}{4}(x-\stackrel{x_1}{1})`

`y-6=4x-4\implies y=4x\underset{\stackrel{\uparrow }{b}}{+2}\qquad \impliedby \begin{array}ll \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}`