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The equation of line cc is y=−7x+10y = -7x + 10. Line dd is perpendicular to line cc and passes through (7,4)(7,4). What is the equation of line dd?\newlineWrite the equation in slope-intercept form.

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Q. The equation of line cc is y=−7x+10y = -7x + 10. Line dd is perpendicular to line cc and passes through (7,4)(7,4). What is the equation of line dd?\newlineWrite the equation in slope-intercept form.
  1. Determine slope of line c: Determine the slope of line c. The equation of line c is given as y=−7x+10y = -7x + 10. 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 c is −7-7.
  2. Find slope of line dd: Find 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 −7-7 is 17\frac{1}{7}. Therefore, the slope of line dd is 17\frac{1}{7}.
  3. Use point-slope form: Use the point-slope form to find the equation of line dd. We have the slope of line dd (17\frac{1}{7}) and a point through which it passes (7,4)(7,4). 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 the point on the line. Plugging in the values, we get (y−4)=17(x−7)(y - 4) = \frac{1}{7}(x - 7).
  4. Simplify to slope-intercept form: Simplify the equation to slope-intercept form.\newlineTo get the slope-intercept form y=mx+by = mx + b, we distribute the slope on the right side of the equation and then isolate yy:\newliney−4=17×x−17×7y - 4 = \frac{1}{7} \times x - \frac{1}{7} \times 7\newliney−4=17×x−1y - 4 = \frac{1}{7} \times x - 1\newliney=17×x−1+4y = \frac{1}{7} \times x - 1 + 4\newliney=17×x+3y = \frac{1}{7} \times x + 3
  5. Verify the equation: Verify the equation.\newlineThe equation y=17×x+3y = \frac{1}{7} \times x + 3 is in slope-intercept form, with a slope of 17\frac{1}{7} and a yy-intercept of 33. This line is perpendicular to line cc, which has a slope of −7-7, and passes through the point (7,4)(7,4), as required.

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