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determine the value of if the points , , and are collinear
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Q. determine the value of if the points , , and are collinear
- Calculate Slope XY: To determine if points X, Y, and Z are collinear, we can use the slope formula. The slope between any two points on a line must be the same if the points are collinear. Let's first find the slope between points X and Y.Slope between X and Y, :
- Calculate Slope YZ: Next, we find the slope between points Y and Z, which should be equal to the slope between X and Y if the points are collinear.Slope between Y and Z, :$m_{YZ} = \(3\)
- Solve for k: Now, we set the slopes equal to each other to solve for \(k\), because the slopes must be the same for the points to be collinear.\[\frac{6}{(k - 3)} = 3\]Cross-multiply to solve for \(k\):\[6 = 3(k - 3)\]\[6 = 3k - 9\]Add \(9\) to both sides:\[6 + 9 = 3k\]\[15 = 3k\]Divide both sides by \(3\):\[\frac{15}{3} = k\]\[5 = k\]
- Verify Solution: We have found the value of \(k\) to be \(5\). To ensure there are no math errors, we can substitute \(k\) back into the slope equations to verify that the slopes are indeed equal.\(\newline\)Substitute \(k = 5\) into \(m_{XY}\):\(\newline\)\(m_{XY} = \frac{6}{(5 - 3)}\)\(\newline\)\(m_{XY} = \frac{6}{2}\)\(\newline\)\(m_{XY} = 3\)\(\newline\)Since \(m_{XY} = m_{YZ} = 3\), the points are collinear, and our value for \(k\) is correct.
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