Problem 11 - Algebra and Trigonometry

Question

Find the slope of the line through P and Q. $$ P(1,-3), Q(-1,6) $$

Step-by-Step Solution

Answer

The slope of the line through P and Q is \(-\frac{9}{2}\).

Step 1: Identify Coordinates

Locate the coordinates of the given points: \(P(1, -3)\) and \(Q(-1, 6)\). This helps in identifying which values correspond to \(x_1, y_1\) and \(x_2, y_2\).

Step 2: Apply Slope Formula

Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) where \((x_1, y_1) = (1, -3)\) and \((x_2, y_2) = (-1, 6)\).

Step 3: Substitute Values

Substitute the coordinates into the slope formula: \( m = \frac{6 - (-3)}{-1 - 1} \).

Step 4: Calculate Numerator

Compute the numerator: \( 6 - (-3) = 6 + 3 = 9 \).

Step 5: Calculate Denominator

Compute the denominator: \(-1 - 1 = -2\).

Step 6: Compute Slope

Divide the calculated numerator by the denominator: \( m = \frac{9}{-2} = -\frac{9}{2} \).

Key Concepts

Slope Formula

The slope of a line is a crucial concept in coordinate geometry, especially when working with linear equations. It measures the steepness or incline of a line, providing insights into how one variable changes with respect to another. To find the slope of a line passing through two distinct points, you can use the slope formula, which is defined as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In this formula, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two points on the line. The numerator \( y_2 - y_1 \) represents the vertical change, or rise, between the two points, while the denominator \( x_2 - x_1 \) represents the horizontal change, or run.

• If the slope \( m \) is positive, the line inclines upwards as it moves from left to right.
• If the slope is negative, the line declines.
• A zero slope indicates a horizontal line, while an undefined slope (caused by a denominator of zero) indicates a vertical line.

Using the slope formula, you can transform raw coordinate data into meaningful analyses of relationships between variables.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to study geometry using a coordinate system. By employing algebraic methods, we can explore geometric problems and relationships in a more analytical way. It combines algebra and geometry by positioning shapes on a coordinate plane.

• The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
• Each point on this plane is represented as an ordered pair \( (x, y) \), where \( x \) is the horizontal displacement from the origin and \( y \) is the vertical displacement.

In our example problem, the points \( P(1, -3) \) and \( Q(-1, 6) \) are used to find the slope of the line connecting them. Coordinate geometry provides a visual approach to understanding slopes as it graphically illustrates the line that passes through these points. By plotting these points on the coordinate plane, you can see the line and its inclination or declination clearly, aiding in a better conceptual understanding.

Linear Equations

Linear equations represent straight lines on a graph, and they can be expressed in various forms such as slope-intercept form \( y = mx + b \), point-slope form, or standard form. In each, the slope \( m \) plays a vital role in defining the characteristics of the line.A linear equation essentially describes a linear relationship between two variables. For the coordinates \( P(1, -3) \) and \( Q(-1, 6) \), the slope \( m = -\frac{9}{2} \) indicates how changes in \( x \) affect the value of \( y \). In real-world applications, such relationships might represent anything from the rate of speed over time to changes in cost with respect to quantity.

• The intercept \( b \) in the slope-intercept form represents the point where the line crosses the y-axis.
• The steeper the slope, the more the line "climbs" or "falls" per unit of x.

Understanding linear equations is fundamental for solving problems involving relationships between two quantitative variables. It allows you to model and predict behaviors within a logical framework, turning raw data into actionable insights.