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The equation for line c c can be written as y=−37x+4 y=-\frac{3}{7}x+4 . Perpendicular to line c c is line d d , which passes through the point (2,4) (2,4) . What is the equation of line d d ? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

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Full solution

Q. The equation for line c c can be written as y=−37x+4 y=-\frac{3}{7}x+4 . Perpendicular to line c c is line d d , which passes through the point (2,4) (2,4) . What is the equation of line d d ? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.
  1. Identify slope of line c: Identify the slope of line c. The equation of line c is given in slope-intercept form as y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. For line c, the slope mm is −37-\frac{3}{7}.
  2. Determine slope of line dd: Determine the slope of line dd. Since line dd is perpendicular to line cc, its slope will be the negative reciprocal of the slope of line cc. The negative reciprocal of −37-\frac{3}{7} is 73\frac{7}{3}.
  3. Use point-slope form: Use the point-slope form to write the equation of line dd. The point-slope form is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is a point on the line. We have the slope 73\frac{7}{3} and the point (2,4)(2,4).
  4. Substitute slope and point: Substitute the slope and point into the point-slope form. This gives us y−4=(73)(x−2)y - 4 = \left(\frac{7}{3}\right)(x - 2).
  5. Distribute slope: Distribute the slope on the right side of the equation. This gives us y−4=(73)x−(73)⋅2y - 4 = \left(\frac{7}{3}\right)x - \left(\frac{7}{3}\right)\cdot 2.
  6. Simplify equation: Simplify the equation by multiplying 73\frac{7}{3} by 22. This gives us y−4=(73)x−143y - 4 = \left(\frac{7}{3}\right)x - \frac{14}{3}.
  7. Add 44 to both sides: Add 44 to both sides of the equation to solve for yy. This gives us y=(73)x−143+4y = \left(\frac{7}{3}\right)x - \frac{14}{3} + 4.
  8. Convert 44 to fraction: Convert 44 to a fraction with the same denominator as 143\frac{14}{3} to combine the terms. This gives us y=(73)x−143+123.y = \left(\frac{7}{3}\right)x - \frac{14}{3} + \frac{12}{3}.
  9. Combine constant terms: Combine the constant terms. This gives us y=73x−23y = \frac{7}{3}x - \frac{2}{3}.

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