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Line tt has an equation of y=43x−2y = \frac{4}{3}x - 2. Line uu is perpendicular to line tt and passes through (−2,2)(-2,2). What is the equation of line uu?\newlineWrite 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 tt has an equation of y=43x−2y = \frac{4}{3}x - 2. Line uu is perpendicular to line tt and passes through (−2,2)(-2,2). What is the equation of line uu?\newlineWrite 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=43x−2y = \frac{4}{3}x - 2. The slope-intercept form of a line is y=mx+by = mx + b, where mm is the slope. Therefore, the slope of line t is 43\frac{4}{3}.
  2. Find slope of line uu: Find the slope of line uu. Since line uu is perpendicular to line tt, the slope of line uu will be the negative reciprocal of the slope of line tt. The negative reciprocal of 43\frac{4}{3} is −34-\frac{3}{4}.
  3. Use point-slope form: Use the point-slope form to find the equation of line uu. Line uu passes through the point (−2,2)(-2,2) and has a slope of −34-\frac{3}{4}. The point-slope form of a line 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 the values, we get y−2=−34(x−(−2))y - 2 = -\frac{3}{4}(x - (-2)).
  4. Simplify equation of line u: Simplify the equation of line u. Expanding the equation from Step 33, we get y−2=−34(x+2)y - 2 = -\frac{3}{4}(x + 2). Distributing the slope, we have y−2=−34x−34(2)y - 2 = -\frac{3}{4}x - \frac{3}{4}(2). Simplifying further, y−2=−34x−32y - 2 = -\frac{3}{4}x - \frac{3}{2}.
  5. Solve for y: Solve for y to put the equation in slope-intercept form.\newlineAdding 22 to both sides of the equation to solve for y, we get y=−34x−32+2y = -\frac{3}{4}x - \frac{3}{2} + 2. To combine the constant terms, we need a common denominator, which is 22. So, 22 is the same as 42\frac{4}{2}, and we have y=−34x−32+42y = -\frac{3}{4}x - \frac{3}{2} + \frac{4}{2}.
  6. Combine constant terms: Combine the constant terms to find the y-intercept. Combining −32-\frac{3}{2} and 42\frac{4}{2}, we get y=−34x+12y = -\frac{3}{4}x + \frac{1}{2}. This is the equation of line uu in slope-intercept form.

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