Problem 52 - 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. $$ \left(-\frac{3}{4},-\frac{7}{4}\right),\left(-1, \frac{5}{2}\right) $$

Step-by-Step Solution

Answer

The slope of the line that passes through the points is \(-2\).

Step 1: Plot the points

Plot the points \(-3/4, -7/4\) and \(-1, 5/2\) on the graph. These coordinates are represented as \((x1, y1)\) and \((x2, y2)\) respectively.

Step 2: Calculate the slope

Use the slope formula, which is \((y2 - y1) / (x2 - x1)\). Substituting the given points into the formula gives \((5/2 - (-7/4)) / (-1 - (-3/4))\). Solving this expression will give the value of the slope. It's important to note that when subtracting a negative number, it's the same as adding a positive number.

Step 3: Check if the slope exists

If \(x1 = x2\), then the line is vertical and the slope is undefined because it would involve dividing by zero. However, in this case, -1 is not equal to -3/4, so a slope can be calculated.

Key Concepts

Plotting Points

Plotting points on a graph is like placing dots on a piece of paper to represent numbers or pairs of numbers. Each point is determined by two numbers, known as coordinates, which tell you exactly where to place the point on a graph. These are typically written as

• \(x\)-coordinate: It tells you the position on the horizontal axis (left or right).
• \(y\)-coordinate: It informs the position on the vertical axis (up or down).

To plot the given points \((-\frac{3}{4}, -\frac{7}{4})\) and \((-1, \frac{5}{2})\), you would first look at the \(x\)-coordinate:
- Move left from the origin by \(\frac{3}{4}\) and \(1\).
Then, check the \(y\)-coordinate:
- Move down by \(\frac{7}{4}\) and up by \(\frac{5}{2}\).
Each pair lands directly on the graph, making it easy to visualize their relationship.

Slope Formula

The slope of a line is a measure of its steepness and is a key value in understanding how two points on a line relate to one another. The slope formula is:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula helps calculate the slope \(m\) by taking the difference in the \(y\)-coordinates (vertical change) and dividing it by the difference in the \(x\)-coordinates (horizontal change).
For our points \((-\frac{3}{4}, -\frac{7}{4})\) and \((-1, \frac{5}{2})\), plug these into the formula:

• \(y_2 = \frac{5}{2}, \, y_1 = -\frac{7}{4}\): \(y_2 - y_1 = \frac{5}{2} + \frac{7}{4}\).
• \(x_2 = -1, \, x_1 = -\frac{3}{4}\): \(x_2 - x_1 = -1 + \frac{3}{4}\).

After performing the arithmetic, you'll obtain the slope of the line connecting these points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate system to interpret geometric figures. It bridges algebra and geometry through graphs of equations and shapes on coordinate planes.
The main elements are:

• The coordinate plane itself, having an \(x\)-axis (horizontal) and a \(y\)-axis (vertical).
• Points, such as our examples \((-\frac{3}{4}, -\frac{7}{4})\) and \((-1, \frac{5}{2})\), which lie on this plane and are denoted by coordinates \((x, y)\).
• Lines, which can be understood by their slopes, intercepts, or equations derived from such points.

Using this knowledge, we can visualize points and lines, solve geometric problems algebraically, and analyze complex shapes and their properties. It's an essential tool in mathematics.

Undefined Slope

An undefined slope occurs when you try to draw a vertical line on a graph. This happens when two points on a line have the same \(x\)-coordinate, meaning they are stacked directly above one another along the vertical axis.
The problem arises because calculating the slope demands division by zero, which isn't possible in algebra:

• For a vertical line with points \((x_1, y_1)\) and \((x_2, y_2)\), if \(x_1 = x_2\), the formula becomes \(\frac{y_2 - y_1}{0}\).
• Division by zero leads to an undefined slope because you can’t determine the exact steepness or direction of the line.

Thus, longer slopes, or more gradual slopes, aren't valid concepts with vertical lines. However, for our problem, since our points have differing \(x\)-values, the slope is defined, reinforcing the fact that learning about different types of slopes is crucial in math.