Problem 119
Question
Consider the line that passes through \(P(-2,3)\) and \(Q(4,-4)\). Find the slope of line \(P Q\)
Step-by-Step Solution
Verified Answer
The slope of line \(PQ\) is \(-\frac{7}{6}\).
1Step 1: Understanding the formula for slope
The slope of a line is calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. In this exercise, these points are \(P(-2, 3)\) and \(Q(4, -4)\).
2Step 2: Substitute the coordinates into the slope formula
Substitute \((x_1, y_1) = (-2, 3)\) and \((x_2, y_2) = (4, -4)\) into the slope formula. This gives us: \[ m = \frac{-4 - 3}{4 - (-2)} \].
3Step 3: Calculate the difference in y-coordinates
Calculate \(-4 - 3\). This results in \(-7\).
4Step 4: Calculate the difference in x-coordinates
Calculate \(4 - (-2)\). This results in \(4 + 2 = 6\).
5Step 5: Calculate the slope
Now, divide the differences calculated: \[ m = \frac{-7}{6} \]. Thus, the slope of the line is \(-\frac{7}{6}\).
Key Concepts
Coordinate GeometrySlope FormulaPoints on a Line
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the properties and position of points on a plane. It allows us to solve geometric problems using algebraic equations.
Using this method, we represent each point on the plane with an ordered pair of numbers, typically called coordinates. Each pair is denoted as \(x, y\), where \(x\) refers to the horizontal position, and \(y\) refers to the vertical position.
In the context of our exercise, we consider two points: \(-2, 3\) and \(4, -4\). These represent P and Q, which are crucial for determining the line's slope.
Coordinate geometry enables us to understand the spatial relationship between points. It helps us calculate distances, angles, and other properties like the slope, which reveals how steep a line is.
Using this method, we represent each point on the plane with an ordered pair of numbers, typically called coordinates. Each pair is denoted as \(x, y\), where \(x\) refers to the horizontal position, and \(y\) refers to the vertical position.
In the context of our exercise, we consider two points: \(-2, 3\) and \(4, -4\). These represent P and Q, which are crucial for determining the line's slope.
Coordinate geometry enables us to understand the spatial relationship between points. It helps us calculate distances, angles, and other properties like the slope, which reveals how steep a line is.
Slope Formula
The slope formula is essential in coordinate geometry when determining the steepness of a line. It's defined as the ratio of the change in vertically to the change horizontally between two distinct points.
Mathematically, the slope (m) is expressed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the given points \((-2, 3)\) and \(4, -4\), we substitute their coordinates into the formula. Here, \(x_1 = -2\), \(y_1 = 3\), \(x_2 = 4\), and \(y_2 = -4\).
Thus, the slope of the line through points P and Q is \(-\frac{7}{6}\). This measure tells us that for every 6 units we move horizontally, the line moves 7 units downward.
Mathematically, the slope (m) is expressed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the given points \((-2, 3)\) and \(4, -4\), we substitute their coordinates into the formula. Here, \(x_1 = -2\), \(y_1 = 3\), \(x_2 = 4\), and \(y_2 = -4\).
- First, find the difference in \(y\)-coordinates: \(-4 - 3 = -7\).
- Next, find the difference in \(x\)-coordinates: \((4 - (-2)) = 6\).
Thus, the slope of the line through points P and Q is \(-\frac{7}{6}\). This measure tells us that for every 6 units we move horizontally, the line moves 7 units downward.
Points on a Line
Points on a line are fundamental in understanding and determining many geometric properties, like the slope and equation of the line. Each point provides a fixed location on the coordinate plane, described by its coordinates.
In the given problem, we use two specific points, P \((-2, 3)\) and Q \(4, -4\), to calculate the line's slope.
Understanding points on a line also helps in determining other line equations and graphing lines on a coordinate plane. The more accurately you understand each point, the better you can visualize and work with the corresponding line.
In the given problem, we use two specific points, P \((-2, 3)\) and Q \(4, -4\), to calculate the line's slope.
- Point P at \((-2, 3)\) indicates that when \(x = -2\), \(y = 3\).
- Point Q at \(4, -4\) shows that when \(x = 4\), \(y = -4\).
Understanding points on a line also helps in determining other line equations and graphing lines on a coordinate plane. The more accurately you understand each point, the better you can visualize and work with the corresponding line.
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