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The equation of line tt is y=74x+2y = \frac{7}{4}x + 2. Perpendicular to line tt is line uu, which passes through the point (2,−2)(2,-2). What is the equation of line uu? 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. The equation of line tt is y=74x+2y = \frac{7}{4}x + 2. Perpendicular to line tt is line uu, which passes through the point (2,−2)(2,-2). What is the equation of line uu? 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 t: Determine the slope of line t.\newlineThe equation of line t is given as y=74x+2y = \frac{7}{4}x + 2. The slope (mm) of a line in the slope-intercept form y=mx+by = mx + b is the coefficient of xx. Therefore, the slope of line t is 74\frac{7}{4}.
  2. Find slope of line u: Find the slope of line u. Since line u is perpendicular to line t, its slope will be the negative reciprocal of the slope of line t. The negative reciprocal of 74\frac{7}{4} is −47-\frac{4}{7}. Therefore, the slope of line u is −47-\frac{4}{7}.
  3. Use point-slope form: Use the point-slope form to find the equation of line uu. We have the slope of line uu −47-\frac{4}{7} and a point through which it passes (2,−2)(2,-2). 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 the point on the line. Plugging in the values, we get y−(−2)=−47(x−2)y - (-2) = -\frac{4}{7}(x - 2).
  4. Simplify to slope-intercept form: Simplify the equation to slope-intercept form.\newlineStarting with y+2=−47(x−2)y + 2 = -\frac{4}{7}(x - 2), distribute the slope −47-\frac{4}{7} across (x−2)(x - 2) to get y+2=−47x+87y + 2 = -\frac{4}{7}x + \frac{8}{7}. Then, subtract 22 from both sides to isolate yy, which gives us y=−47x+87−147y = -\frac{4}{7}x + \frac{8}{7} - \frac{14}{7}. Simplifying the y-intercept, we get y=−47x−67y = -\frac{4}{7}x - \frac{6}{7}.

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