Problem 38 - Elementary Algebra

Question

Sketch the line determined by each pair of points and decide whether the slope of the line is positive, negative, or zero. $$(-3,-4),(5,-4)$$

Step-by-Step Solution

Answer

The slope of the line is zero; it is a horizontal line.

Step 1: Understand the Problem

We are given two points, (-3, -4) and (5, -4). The task is to sketch the line that passes through these points and determine the slope of the line.

Step 2: Recall the Slope Formula

The formula for the slope, usually denoted as \(m\), of a line passing through 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 3: Substitute Values into Slope Formula

Substitute the coordinates of the points into the slope formula: \[ m = \frac{-4 - (-4)}{5 - (-3)} = \frac{-4 + 4}{5 + 3} = \frac{0}{8} \]

Step 4: Simplify the Result

The calculation \( \frac{0}{8} \) simplifies to 0. This means the slope \(m = 0\).

Step 5: Interpret the Slope

A slope of 0 indicates that the line is horizontal.

Step 6: Conclusion

Since the line is horizontal, it spans from (-3, -4) to (5, -4) without vertical change. Therefore, the slope of the line is 0 (zero).

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is a fundamental skill in algebra. A linear equation represents a straight line when graphically plotted on a coordinate plane. To draw this line, one must have at least two points through which it passes.

• The equation of a line can often be given in slope-intercept form: \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
• Alternatively, finding two points on the line, as given in the exercise, allows us to draw a line by plotting these points and connecting them with a straight line.
• After finding the slope from these points, you can better understand the line's direction and steepness.

In our problem, the points \((-3, -4)\) and \((5, -4)\) lie on the same horizontal line due to their identical y-values, making it straightforward to sketch.

Horizontal Line

A horizontal line is one of the simplest forms of a line in geometry. It maintains a constant y-value across different x-values, meaning there is no vertical change as you move along the line.

• If a line is horizontal, it always has a slope (\(m\)) of zero.
• Its graph is parallel to the x-axis, resulting in a straight line that does not ascend or descend.
• For an equation in the form of \(y = b\), every point on the line shares the same y-coordinate, which is \(b\). In our step-by-step example, \((-3, -4)\) and \((5, -4)\) both have \(y = -4\).

Recognizing a horizontal line quickly helps determine its slope and graph it easily, without complex calculations.

Slope Formula

The slope formula provides a way to measure how steep a line is by comparing the vertical change to the horizontal change between two points. The formula for calculating the slope \(m\) is\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• The values \((x_1, y_1)\) and \((x_2, y_2)\) represent two distinct points on a line.
• The numerator \(y_2 - y_1\) measures the vertical difference between the points, while the denominator \(x_2 - x_1\) measures the horizontal difference.
• A zero in the numerator indicates no vertical change, leading to a slope of zero, as seen in the example exercise.

Understanding the slope formula allows students to determine the steepness and direction of a line; whether it's flat, ascending, or descending.