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Line kk has an equation of y=x+27y = x + \frac{2}{7}. Line ℓ\ell includes the point (7,−2)(7,-2) and is perpendicular to line kk. What is the equation of line ℓ\ell? 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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Q. Line kk has an equation of y=x+27y = x + \frac{2}{7}. Line ℓ\ell includes the point (7,−2)(7,-2) and is perpendicular to line kk. What is the equation of line ℓ\ell? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.
  1. Determine slope of line: Determine the slope of line kk.\newlineLine kk has the equation y=x+27y = x + \frac{2}{7}. This is in slope-intercept form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. Therefore, the slope of line kk is 11.
  2. Find perpendicular slope: Find the slope of the line that is perpendicular to line kk. The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Since the slope of line kk is 11, the negative reciprocal is −1-1.
  3. Use point-slope form: Use the point-slope form to find the equation of the line.\newlineWe have a point (7,−2)(7, -2) and a slope −1-1. The point-slope form of a line's equation 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. Plugging in our values, we get y−(−2)=−1(x−7)y - (-2) = -1(x - 7).
  4. Simplify to slope-intercept form: Simplify the equation from point-slope form to slope-intercept form.\newlineStarting with y+2=−1(x−7)y + 2 = -1(x - 7), we distribute the −1-1 to get y+2=−x+7y + 2 = -x + 7. Then, we subtract 22 from both sides to isolate yy, resulting in y=−x+5y = -x + 5.
  5. Check for errors: Check the equation for any mathematical errors.\newlineThe equation y=−x+5y = -x + 5 is in slope-intercept form, with a slope of −1-1 and a y-intercept of 55. There are no mathematical errors in the simplification process.

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