Problem 69 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the ordered pairs \((-2,2),(0,0),\) and \((2,2)\) to graph a straight line.

Step-by-Step Solution

Answer

No, the statement does not make sense as the slopes between the pairs of points are not equal. Hence, these points cannot graph a straight line.

Step 1: Understanding the Concept of a Straight Line

The fundamental characteristic of a straight line in a two-dimensional coordinate plane is that the slope between any two points on the line is always constant. In other words, if we select any two points on the line, the ratio between the vertical change (rise) and the horizontal change (run) between them always remains the same.

Step 2: Calculating the Slopes

Let's calculate the slope between the first and second point, and then the slope between the second and third point. To calculate the slope, we use the formula \((y_{2}-y_{1})/(x_{2}-x_{1})\). \nFor the first pair, \((y_{2}-y_{1})/(x_{2}-x_{1}) = (0-2)/ (0 - (-2)) = -1\). Now, let's compute the slope for the second pair of points, \((y_{2}-y_{1})/(x_{2}-x_{1}) = (2-0)/(2-0) = 1\).

Step 3: Comparing the Slopes

Comparing the slopes determined from Steps 2, we can see that the slope between the first pair of points (-1) is not equal to the slope between the second pair of points (1). Therefore, these three points can't be connected by a straight line.