Question

Determine whether the given points lie on a straight line. $$ A(-3,6), B(3,3), \text { and } C(6,0) $$

Short answer

The slopes between points A and B, and B and C are \(m_{AB} = -\frac{1}{2}\) and \(m_{BC} = -1\), respectively. Since they are not equal, the points A(-3, 6), B(3, 3), and C(6, 0) do not lie on a straight line.

Step-by-step solution

Compute the slope between A and B

First, let's find the slope between points A(-3, 6) and B(3, 3). We need to apply the formula: \[m_{AB} = \frac{y2 - y1}{x2 - x1} = \frac{3-6}{3-(-3)}\]

Calculate the slope between A and B

Next, substitute the coordinates of points A and B into the slope formula and calculate the result: \[m_{AB} = \frac{-3}{6} = -\frac{1}{2}\]

Compute the slope between B and C

Now, let's find the slope between points B(3, 3) and C(6, 0). We need to apply the same formula as before: \[m_{BC} = \frac{y2 - y1}{x2 - x1} = \frac{0 - 3}{6 - 3}\]

Calculate the slope between B and C

Substitute the coordinates of points B and C into the slope formula and calculate the result: \[m_{BC} = \frac{-3}{3} = -1\]

Compare the slopes

Now we compare the slopes m_{AB} and m_{BC}: \[m_{AB} = -\frac{1}{2} \neq -1 = m_{BC}\] Since the slopes are not equal, the points A(-3, 6), B(3, 3), and C(6, 0) do not lie on a straight line.