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Chapter 1: Problem 68

Determine whether the points lie on a straight line. $$ A(-1,7), B(2,-2), \text { and } C(5,-9) $$

Short Answer

Expert verified
The slopes between points A and B, and B and C are -3 and \(\frac{-7}{3}\), respectively. Since the slopes are not equal, the points A(-1,7), B(2,-2), and C(5,-9) do not lie on a straight line.

Step by step solution

01

Calculate the slope between points A and B

We can find the slope between two points using the formula: \(m = \frac{y2 - y1}{x2 - x1}\) For points A(-1,7) and B(2,-2), let's plug in the coordinates into the slope formula: \(m_{AB} = \frac{-2 - 7}{2 - (-1)}\)
02

Simplify the slope between points A and B

Now let's simplify the slope: \(m_{AB} = \frac{-2 - 7}{2 - (-1)} = \frac{-9}{3}\) \(m_{AB} = -3\) So the slope between points A and B is -3.
03

Calculate the slope between points B and C

Next, we'll find the slope between points B(2,-2) and C(5,-9) using the slope formula: \(m_{BC} = \frac{-9 - (-2)}{5 - 2}\)
04

Simplify the slope between points B and C

Now let's simplify the slope: \(m_{BC} = \frac{-9 - (-2)}{5 - 2} = \frac{-7}{3}\) So the slope between points B and C is \(\frac{-7}{3}\).
05

Compare the slopes

Now that we have both slopes, we can compare them: \(m_{AB}\) = -3 \(m_{BC}\) = \(\frac{-7}{3}\) Since the slopes are not equal, the points A, B, and C do not lie on a straight line.

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