Problem 8 - Algebra and Trigonometry

Question

Find the slope of the line through \(P\) and \(Q .\) \(P(-1,-4), Q(6,0)\)

Step-by-Step Solution

Answer

The slope of the line through points P and Q is \(\frac{4}{7}\).

Step 1: Identifying the Formula

To find the slope of the line through two points, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the points.

Step 2: Substitute Coordinates

Substitute the coordinates of points \(P(-1, -4)\) and \(Q(6, 0)\) into the slope formula. This gives us \( m = \frac{0 - (-4)}{6 - (-1)} \).

Step 3: Simplify the Expression

Simplify the expression \( m = \frac{0 - (-4)}{6 - (-1)} \) to find the slope. Simplifying the numerator and the denominator, we get \( m = \frac{0 + 4}{6 + 1} \).

Step 4: Calculate the Slope

Perform the arithmetic operations: \( m = \frac{4}{7} \). The slope of the line through points \(P\) and \(Q\) is \( \frac{4}{7} \).

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry is a branch of mathematics that combines algebra and geometry to study points, lines, and shapes on a plane. In this framework, points are described using ordered pairs

• X-coordinate: Represents the horizontal distance from the origin.
• Y-coordinate: Represents the vertical distance from the origin.

Each point on the Cartesian plane is defined by these coordinates, such as

• For point \(P\), we have the coordinates \((-1, -4)\)
• For point \(Q\), the coordinates are \((6, 0)\)

When dealing with two points, such as these located on a plane, we can determine the relationship between them using concepts like distance, midpoint, and slope. The slope is particularly useful as it measures the steepness or incline of the line connecting these two points.

Formula for Slope

When asked to find the slope of a line between two points, we use the formula for slope, denoted by \(m\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In this formula:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \(y_2 - y_1\) represents the vertical change (rise).
• \(x_2 - x_1\) represents the horizontal change (run).

Substitute the values of the given points:

• Point \(P(-1, -4)\) means \(x_1 = -1\) and \(y_1 = -4\).
• Point \(Q(6, 0)\) means \(x_2 = 6\) and \(y_2 = 0\).

Thus, the slope can be derived by plugging these into our formula, resulting in: \[ m = \frac{0 - (-4)}{6 - (-1)} \]
The formula accounts for how far and in which direction the line inclines as it moves from one point to another.

Simplifying Arithmetic Expressions

Arithmetic simplification is the process of making expressions easier to work with by performing operations and reducing complexity. For the slope formula, simplification is critical to get the answer.
From the expression obtained: \[ m = \frac{0 - (-4)}{6 - (-1)} \]

• Firstly, recognize that subtracting negative numbers is equivalent to adding their positive counterparts.
• So, \(0 - (-4)\) becomes \(0 + 4\), which simplifies to \(4\).
• Similarly, \(6 - (-1)\) becomes \(6 + 1\), which simplifies to \(7\).

Hence, our simplified expression is \[ m = \frac{4}{7} \]Thus, after performing the arithmetic operations accurately, you find the slope of the line through points \(P\) and \(Q\) is \(\frac{4}{7}\). This step is about clear calculation, ensuring that we end up with the correct value for the slope of the given line.