Question

write an equation (a) in slope-intercept form and (b) in standard form for the line passing through \\((2,7)\\) and perpendicular to \\(3x + 5y = 1\\).

part 1 of 2

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.)

Explanation:

⚡ Using what you learned: parallel and perpendicular lines_equations

Step 1: Find the slope of the given line

Find the slope of the line \(3x + 5y = 1\) by rewriting it in slope-intercept form \(y = mx + b\):

$$ 5y = -3x + 1 $$

$$ y = -\frac{3}{5}x + \frac{1}{5} $$

The slope of the given line is \(m_1 = -\frac{3}{5}\).

Step 2: Determine the perpendicular slope

The slope of a line perpendicular to one with slope \(m_1\) is the negative reciprocal, \(m_2 = -\frac{1}{m_1}\):

$$ m_2 = \frac{5}{3} $$

Step 3: Write the equation in slope-intercept form

Use the point-slope formula with the point \((2,7)\) and slope \(m = \frac{5}{3}\):

$$ y - y_1 = m(x - x_1) $$

$$ y - 7 = \frac{5}{3}(x - 2) $$

$$ y - 7 = \frac{5}{3}x - \frac{10}{3} $$

$$ y = \frac{5}{3}x - \frac{10}{3} + \frac{21}{3} $$

$$ y = \frac{5}{3}x + \frac{11}{3} $$

Step 4: Write the equation in standard form

Convert the slope-intercept equation to standard form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers and \(A \ge 0\):

Multiply the entire equation by 3 to clear the fractions:

$$ 3y = 5x + 11 $$

Subtract \(5x\) from both sides:

$$ -5x + 3y = 11 $$

Multiply by \(-1\) to make the \(x\)-coefficient positive:

$$ 5x - 3y = -11 $$

Answer:

a) The equation of the line in slope-intercept form is \(y = \frac{5}{3}x + \frac{11}{3}\).

b) The equation of the line in standard form is \(5x - 3y = -11\).