Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through $(2,7)$ and perpendicular to $3 x+5 y=1$.
a) The equation of the line in slope-intercept form is
(Type your answer in slope-intercept form. Use integers or fractions for any numbers in the equation.)
Answer
\(\boxed{y = \frac{5}{3}x + \frac{11}{3}}\) is the equation in slope-intercept form and \(\boxed{5x - 3y = -11}\) is the equation in standard form
Step 1 ：Convert the given line $3x + 5y = 1$ to slope-intercept form: $y = -\frac{3}{5}x + \frac{1}{5}$
Step 2 ：Find the slope of the line perpendicular to the given line: $m = \frac{5}{3}$
Step 3 ：Use the point-slope form of a line with point $(2, 7)$ and slope $\frac{5}{3}$: $y - 7 = \frac{5}{3}(x - 2)$
Step 4 ：Convert the equation to slope-intercept form: $y = \frac{5}{3}x + \frac{11}{3}$
Step 5 ：Convert the equation to standard form: $5x - 3y = -11$
Step 6 ：\(\boxed{y = \frac{5}{3}x + \frac{11}{3}}\) is the equation in slope-intercept form and \(\boxed{5x - 3y = -11}\) is the equation in standard form
