Problem 51 - Elementary Algebra Within Reach

Question

In Exercises 49-54, plot the points and find the slope (if possible) of the line that passes through the points. If not possible, state why. $$ \left(\frac{7}{2}, \frac{9}{2}\right),\left(\frac{4}{3},-3\right) $$

Step-by-Step Solution

Answer

The slope of the line passing through the given points is found by plugging the coordinates of the two points into the slope formula, which gives the result \[ m = \frac {-3 - 4.5} {1.33 - 3.5} \]

Step 1: Define the Points

The two points given in the question are: Point A = \(\left(\frac{7}{2}, \frac{9}{2}\right)\), points B = \(\left(\frac{4}{3},-3\right)\). These points can be written in decimal form for ease of plotting on the graph. Therefore, Point A is (3.5, 4.5) and the point B is (1.33, -3). Note that (x1, y1) are coordinates of the point A and (x2, y2) are coordinates of the point B.

Step 2: Plot the Points

Plot these points on a two-dimensional graph. Point A (3.5, 4.5) and point B (1.33, -3) are marked on the graph.

Step 3: Find the Slope

The formula for the slope, denoted by m, of a line passing through points (x1, y1) and (x2, y2) is given as \[ m = \frac {y2 - y1} {x2 - x1} \] Using the coordinates of point A and B in this formula we get: \[ m = \frac {-3 - 4.5} {1.33 - 3.5} \] Divide the numbers to get the slope m.

Key Concepts

Plotting Points on a Graph

Understanding how to plot points on a graph is fundamental in mathematics. A graph consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Here is how you can plot points:

• First, identify the x-coordinate and the y-coordinate of the point you want to plot.
• The x-coordinate tells how far to move horizontally from the origin, while the y-coordinate shows how far to move vertically.
• To plot a point, start at the origin (0,0), move right or left to the x-coordinate, and then up or down to the y-coordinate.

For example, consider the point A \((3.5, 4.5)\):

• Start at the origin.
• Move 3.5 units to the right along the x-axis.
• Then, move 4.5 units up along the y-axis.
• Mark this position as point A.

Repeat these steps for any other points you need to plot. In our current exercise, we also need to plot point B \((1.33, -3)\) similarly. By placing these points on a graph, you can easily visualize the relationship between them.

Coordinates

Coordinates are a pair of numbers used to locate a point on a graph. They give us a position in a two-dimensional space, defined by numerical values.

• The first number is the x-coordinate (or abscissa), representing the position along the horizontal axis.
• The second number is the y-coordinate (or ordinate), representing the position along the vertical axis.

For example, a coordinate such as \((3.5, 4.5)\) tells us:

• The point is 3.5 units along the x-axis.
• The point is 4.5 units up the y-axis.

Similarly, the coordinate \((1.33, -3)\) tells us:

• The point is 1.33 units along the x-axis.
• The point is 3 units down the y-axis.

Understanding these coordinates allows for accurately plotting and visualizing points on a grid. This foundation is crucial for further determining other properties, such as the slope of a line passing through these points.

Finding Slope Formula

Finding the slope of a line connecting two points involves using a specific formula that relates the change in y-coordinates to the change in x-coordinates between the points. The slope, often represented by the letter \(m\), helps describe the steepness and direction of the line.

• The formula to calculate the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• A positive slope indicates that the line is rising as it moves from left to right, while a negative slope shows that the line is falling.
• If the slope is zero, the line is horizontal, and if undefined (division by zero), the line is vertical.

Applying this formula with our example points, point A \((3.5, 4.5)\) and point B \((1.33, -3)\), we calculate:

• \(m = \frac{-3 - 4.5}{1.33 - 3.5}\)
• By performing these calculations, you can determine the exact slope value, which provides insight into how these points are spatially related.

Understanding this formula is vital for interpreting various mathematical and real-world relationships represented graphically.