The equation of line k is y+10= 3(x+3). Perpendicular to line k is line ℓ, which passes through the point (5,-5). What is the equation of line ℓ ? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Question

The equation of line  k is  y+10=  3(x+3). Perpendicular to line  k is line  ℓ, which passes through the point  (5,-5). What is the equation of line  ℓ ? 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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Determine Slope of Line k: Determine the slope of line k. The equation of line k is given in point-slope form: y + 10 = 3(x + 3). To find the slope, we need to rewrite it in slope-intercept form (y = mx + b), where m is the slope. y + 10 = 3x + 9 Subtract 10 from both sides to isolate y. y = 3x + 9 - 10 y = 3x - 1 The slope of line k is 3.Find Slope of Line ℓ: Find the slope of line ℓ, which is perpendicular to line k. Slopes of perpendicular lines are opposite reciprocals. The slope of line k is 3, so the slope of line ℓ will be the negative reciprocal of 3. The negative reciprocal of 3 is -1/3. Therefore, the slope of line ℓ is -1/3.Use Point and Slope to Find Equation: Use the point (5, -5) and the slope -1/3 to find the equation of line ℓ. We will use the point-slope form of the equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Plugging in the point (5, -5) and the slope -1/3, we get: y - (-5) = -1/3(x - 5) y + 5 = -1/3x + 5/3Solve for y in Slope-Intercept Form: Solve for y to put the equation in slope-intercept form. Subtract 5 from both sides to isolate y. y = -1/3x + 5/3 - 5 To combine the terms, we need a common denominator, which is 3. y = -1/3x + 5/3 - 15/3 y = -1/3x - 10/3 Now we have the equation of line ℓ in slope-intercept form.

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