Problem 35 - 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. $$(-1,3),(-6,-2)$$

Step-by-Step Solution

Answer

The slope is positive.

Step 1: Identify the Coordinates

First, we need to identify the coordinates of the two points given in the problem. The points are \((-1, 3)\) and \((-6, -2)\).

Step 2: Calculate the Slope

The formula for the slope \(m\) of the line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \[m = \frac{y_2 - y_1}{x_2 - x_1}\].Substitute the given points into the formula:\[m = \frac{-2 - 3}{-6 - (-1)} = \frac{-5}{-5} = 1.\] So, the slope of the line is \(1\).

Step 3: Determine the Slope's Sign

Since the slope \(m = 1\) is positive, the line will ascend from left to right. Thus, the slope of the line is positive.

Step 4: Visualize the Line

If we were to sketch the line based on the points \((-1, 3)\) and \((-6, -2)\), we would see that it rises as we move from the left point at \((-6, -2)\) to the right point at \((-1, 3)\). This confirms our calculation that the slope is positive.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to use algebra to study geometric concepts on a plane. It essentially means describing geometric figures like lines and shapes in terms of coordinates. These coordinates are the \(x\)-axis and \(y\)-axis values that pinpoint a location on the plane.

When you look at a pair of coordinates such as \((-1, 3)\) or \((-6, -2)\), the first number represents the horizontal position (x-coordinate), and the second number represents the vertical position (y-coordinate). This system allows us to easily plot various points and lines on a graph.

• The origin is the point \( (0, 0) \) where both axes meet.
• The x-axis runs horizontally, while the y-axis runs vertically.
• Positive x-values are to the right of the origin, and negative x-values are to the left.
• Positive y-values are above the origin, and negative y-values are below.

Knowing these basics helps you understand how to plot points and lines and is fundamental when utilizing coordinates for graph-related calculations.

Calculating Slope

The slope of a line is a measure of its steepness and direction. In coordinate geometry, we calculate the slope using two points on the line. The slope is given by the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Here, \(m\) represents the slope, while \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points on the line. The formula effectively determines how much the line rises (changes in the y-direction) for a unit change in the x-direction, often referred to as "rise over run."

• If the slope is positive, as in the solution where \(m = 1\), the line rises from the left to the right.
• If the slope is negative, the line falls from left to right.
• If the slope is zero, the line is horizontal.

Knowing how to calculate the slope is crucial for understanding the behavior of lines in the plane and helps in graphing linear equations and comparing the steepness of different lines.

Graphing Linear Equations

One of the essential skills in coordinate geometry is graphing linear equations. Linear equations often take the form \(y = mx + b\), where \(m\) stands for the slope and \(b\) is the y-intercept, or the point where the line crosses the y-axis.

Let’s look at how you can plot a line given two points, such as \((-1, 3)\) and \((-6, -2)\). Start by plotting these two points on a coordinate plane. Then, draw a straight line through them. This line is the visual representation of the equation that would describe these coordinates.

• The slope, calculated earlier, tells us the tilt of the line.
• Using the slope, you can identify how the line extends across the plane; for every unit right you move on the x-axis, the line goes up by the slope’s value on the y-axis.
• The intercept \(b\) in \(y = mx + b\) would be another step to fully define the line, but it isn’t necessary if you are only comparing points.

By graphing, you can verify the properties of a line, such as its slope and positioning, which offers a visual understanding of equations and the relationships between mathematical concepts.