Problem 31 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \((-3, -2)\), \((1, 6)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (-3, -2) and (1, 6) is 2

Step 1: Identify the Points

The points given are (-3, -2) and (1, 6). We can let (-3, -2) be (x1, y1) and (1, 6) be (x2, y2)

Step 2: Substitute into the Slope Formula

Plugging these coordinates into our formula, we get m = (6 + 2) / (1 + 3)

Step 3: Simplify the calculation

Solving inside the brackets we get m = 8 / 4

Step 4: Calculating the Slope

Finally, simplifying the expression gives us m = 2

Key Concepts

Plotting Points

To accurately represent the location of a point in a two-dimensional space, we utilize a system called the coordinate system. Imagine it as a map with two intersecting lines, one horizontal (the x-axis) and one vertical (the y-axis). Each point is like a specific address on this map, determined by an x (horizontal) value and a y (vertical) value.

For the points (-3, -2) and (1, 6), you start at the origin, where the x-axis and the y-axis meet. For (-3, -2), move 3 units to the left (since it's a negative x value) and 2 units down (negative y value). For (1, 6), you'd go 1 unit right and 6 units up. This plotting process lays the foundation for visualizing and solving many types of mathematical problems, such as finding the slope of a line.

Slope Formula

The slope of a line is a measure of its steepness and can be calculated using the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). In this formula, \(m\) is the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two distinct points on the line. The slope tells us how much the y-value changes (rises or falls) for each unit of change in the x-value.

Using our example, plugging the coordinates (-3, -2) and (1, 6) into the formula will give us the slope of the line that passes through these two points. This calculation is an essential step for graphing linear equations and understanding the direction and angle of lines in a coordinate space.

Coordinate System

The coordinate system is a fundamental concept for graphing and navigating through mathematics. It consists of two perpendicular lines called axes, which intersect at the origin, marked as \( (0, 0) \). Each point in the plane is determined by an ordered pair of numbers, \((x, y)\), with \(x\) representing the horizontal position and \(y\) the vertical position.

Understanding how to use this system is vital for multiple disciplines of math, including algebra and geometry. Mastery of the coordinate system enables students to visualize equations, inequalities, and functions. It also plays a crucial role in more advanced fields like calculus and is used in real-world applications such as GPS technology and computer graphics.

Simplifying Expressions

Simplifying expressions is like tidying up a room—reducing clutter and organizing things to make them easier to understand. In algebra, this means combining like terms, factoring, expanding expressions, and performing arithmetic operations to make expressions as straightforward as possible.

For instance, in our slope calculation, simplifying the expression \( m = \frac{8}{4} \) is crucial, as it gives us the precise value of the slope, \( m = 2 \). Simple expressions allow us to better analyze and interpret the problem and create a clearer path to solving it. It’s a core skill in mathematics that helps simplify complex problems and find most efficient solutions.