Question - Equation of Perpendicular Line
Solution:
The equation of line p is given as y = -4x + 1. This equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. For line p, the slope is -4.Line q is supposed to be perpendicular to line p. When two lines are perpendicular, the product of their slopes is -1. This means that if the slope of line p is m, then the slope of line q will be -1/m. Therefore, the slope of line q is -1/(-4) = 1/4.Now that we know the slope of line q is 1/4, we can use the point it passes through, (-6, 1), to find the y-intercept (b) of line q.Starting with the point-slope form of the line equation:y - y1 = m(x - x1)Plugging in the slope (m = 1/4) and the point (-6, 1):y - 1 = 1/4(x - (-6))y - 1 = 1/4(x + 6)Now, distribute 1/4 to (x + 6):y - 1 = 1/4x + 1/4(6)Simplify:y - 1 = 1/4x + 6/4y - 1 = 1/4x + 3/2Finally, we want to write this in slope-intercept form, so we solve for y by adding 1 to both sides:y = 1/4x + 3/2 + 1Since we want to write numbers as simplified fractions or integers, let's convert 1 to a fraction with a denominator of 2:y = 1/4x + 3/2 + 2/2Combine the fractions:y = 1/4x + 5/2And so, the equation of line q in slope-intercept form is:y = 1/4x + 5/2
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