Use a determinant to find y such that (8,-20), (16,y),and (20,-8)are collinear.
A)y = -12
B)y = 20
C)y = -44
D)y = 4
E)y = -8

Question

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For three points to be collinear, the determinant of the matrix formed by their coordinates (with each point extended by a 1 as the third coordinate) must be zero. The matrix and its determinant for points (8, -20, 1), (16, y, 1), and (20, -8, 1) is:| 8 -20 1 || 16 y 1  || 20 -8 1 |The determinant is 8(y+1) - 20(16-20) + 1(16*(-8) - (-20)*20) = 8y + 8 + 80 - 320 + 400 = 8y + 168.Setting the determinant to 0 for collinearity:8y + 168 = 08y = -168y = -21However, there seems to be a mistake in my calculation as -21 is not an option and does not match the correct answer. Let's correct the calculation of the determinant:The correct determinant calculation for points to be collinear is:| 8 -20 1 || 16 y 1  || 20 -8 1 |Determinant = 8(y-1) - 16(-20 + 8) + (160 + 20y) = 8y - 8 + 192 + 160 + 20y = 28y + 344.For the points to be collinear, the determinant should be 0, which means my recalculation has an error. The correct approach to find the determinant for collinearity is:Determinant = 8(y+1) - 20(16-20) + 1(16*(-8) - (-20)*20) does not lead to the correct setup for this problem. Instead, the determinant of the matrix formed by the coordinates should directly lead to a condition that can be solved for y. Given the mistake in the recalculation, let's directly apply the formula for the determinant of three points (x1, y1), (x2, y2), (x3, y3) being collinear:0 = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)Plugging the points directly into this formula or using the determinant method correctly should give us the correct answer without the need for recalculation. The correct determinant formula for these points is:0 = 8(y + 8) - 20(16 - 20) + 16(-20 + 8)Given the options and realizing there was a mistake in the determinant calculation process, the correct approach involves setting up the determinant correctly from the start and solving for y. The correct answer should match one of the provided options based on the correct determinant calculation for collinearity.

Use a Determinant to Find Y Such That (8,-20), (16,y),and