Problem 10 - Elementary Algebra

Question

For the following two problems, find the slope, if it exists, of the line containing the following points. $$ (-6,-1) \text { and }(0,8) $$

Step-by-Step Solution

Answer

Answer: The slope of the line passing through the points (-6, -1) and (0, 8) is $\frac{3}{2}$.

Step 1: Identify the given points

We are given the points \((-6, -1)\) and \((0, 8)\). Let's call \((-6, -1)\) as point \(A\) and \((0, 8)\) as point \(B\). We have \(A(x_1, y_1) = (-6, -1)\) and \(B(x_2, y_2) = (0, 8)\).

Step 2: Apply the slope formula

We will use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) to calculate the slope of the line containing points \(A\) and \(B\). Plugging in the values for \(x_1\), \(y_1\), \(x_2\), and \(y_2\), we get: $$ m = \frac{8 - (-1)}{0 - (-6)} $$

Step 3: Simplify the expression

Now let's simplify the expression: $$ m = \frac{8 + 1}{0 + 6} = \frac{9}{6} $$

Step 4: Reduce the fraction

We can further reduce the fraction \(\frac{9}{6}\) to its simplest form: $$ m = \frac{3}{2} $$ So the slope of the line containing the points \((-6, -1)\) and \((0, 8)\) is \(\frac{3}{2}\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of geometry where the position of points on a plane is described using an ordered pair of numbers known as coordinates. This system, often called the Cartesian coordinate system, allows you to determine geometric features using algebraic equations. When you use points like \(-6, -1\) and \(0, 8\), you can define a line that connects these points on a graph. Each point is represented by \( (x, y) \), where \(x\) is the horizontal position and \(y\) is the vertical position of the point.

Knowing the coordinates of points can help us find various features of a line like its slope, direction, and length. Here are some key terms frequently used:

• Intercepts: Points where the line crosses the axes.
• Distance: The length between two points.
• Slope: The steepness or incline of the line.

Using coordinate geometry, you can turn geometric problems into algebraic equations, making them easier to solve.

Slope Formula

The slope of a line is a measure of its steepness and direction. It’s calculated from the change in the vertical direction (rise) compared to the change in the horizontal direction (run) between two points. This is described using the slope formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points on the line.

In our exercise, you use the formula with points \((-6, -1)\) and \((0, 8)\):

• Substitute values: \(m = \frac{8 - (-1)}{0 - (-6)}\)
• Calculate vertical change: \(8 - (-1) = 9\)
• Calculate horizontal change: \(0 - (-6) = 6\)
• So the slope \(m = \frac{9}{6}\)

Understanding the slope helps in knowing how a line is oriented on a plane, helping in predicting and analyzing trends when representing data graphically.

Reducing Fractions

Reducing fractions is the process of simplifying a fraction to its lowest terms. This makes it easier to work with or interpret the results. A fraction is reduced by dividing both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).

In our exercise, the slope was initially calculated as \(\frac{9}{6}\). To simplify this:

• Identify the GCD of 9 and 6, which is 3.
• Divide the numerator and denominator each by 3: \(\frac{9}{3} = 3\) and \(\frac{6}{3} = 2\).
• Thus, \(\frac{9}{6}\) simplifies to \(\frac{3}{2}\), which is the reduced or simplified form of the slope.

Reducing fractions helps in making solutions neater and easier to understand, especially in geometric interpretations and whenever the computed quantity involves fractions.