Problem 73 - Understanding Elementary Algebra with Geometry

Question

How could you use the idea of slope to show that the three points \((-1,-2)\) \((2,0),\) and \((5,2)\) all lie on a straight line?

Step-by-Step Solution

Answer

The slopes between all pairs of points are equal (\frac{2}{3}), so the points lie on a straight line.

Step 1 - Define the Slope Formula

Use the slope formula to determine if the slopes between the pairs of points are equal. The slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 2 - Calculate Slope Between First Pair

Find the slope between the points \( (-1, -2) \) and \( (2, 0) \) using the formula: \[ m_{12} = \frac{0 - (-2)}{2 - (-1)} = \frac{2}{3} \]

Step 3 - Calculate Slope Between Second Pair

Find the slope between the points \( (2, 0) \) and \( (5, 2) \) using the formula: \[ m_{23} = \frac{2 - 0}{5 - 2} = \frac{2}{3} \]

Step 4 - Compare the Slopes

Compare the slopes obtained in Steps 2 and 3. If the slopes are equal, then the points lie on a straight line. \[ m_{12} = m_{23} = \frac{2}{3} \]

Step 5 - Verify with Third Pair

For thoroughness, you can also find the slope between the first and third points \( (-1, -2) \) and \( (5, 2) \). \[ m_{13} = \frac{2 - (-2)}{5 - (-1)} = \frac{4}{6} = \frac{2}{3} \]

Step 6 - Conclude

Since all calculated slopes are equal \( m_{12} = m_{23} = m_{13} = \frac{2}{3} \), the points \( (-1, -2) \), \( (2, 0) \), and \( (5, 2) \) all lie on a straight line.

Key Concepts

headline of the respective core concept

To get a deep understanding of the slope, let's first define the 'slope formula.' Slope is essentially a measure of the steepness or incline of a line. In coordinate geometry, it helps us understand how two points are related on a graph.

headline of the respective core concept

The slope formula is crucial for calculating slope. It is given by the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\) '
• This formula calculates the slope \((m)\) between two points
  • (x1, y1) and (x2, y2).
  • The difference \((y_2 - y_1)\) represents the rise, or the vertical change.
  • The difference \((x_2 - x_1)\) stands for the run, or the horizontal change.
  Slope shows how much the y-value changes for a change in the x-value. Heights and levels in the different coordinates are determined in accordance with how steep the slope is, with steepness dictated by the change in y divided by the change in x.

headline of the respective core concept

Calculating slope involves applying the slope formula to specific pairs of points and comparing the results. Let's look at the given problem: All points should ideally have the same slope when we compare pairs.

headline of the respective core concept

Coordinate geometry is the branch of mathematics that uses coordinates to study the relationships between points. In the given exercise, the slope formula helps us determine if points are collinear (meaning they lie on the same straight line).