Problem 5 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. (-1,4) \text { and }(3,-2)

Step-by-Step Solution

Answer

The slope is -1.5.

Step 1 - Recall the Slope Formula

The formula for the slope of a line passing through two points \text{A}(x_1, y_1) and \text{B}(x_2, y_2) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, the slope is denoted by m and requires the coordinates of the two points.

Step 2 - Identify the Coordinates

Identify the coordinates of the two given points. Let \text{A} = (-1, 4) and \text{B} = (3, -2), where \( x_1 = -1 \), \( y_1 = 4 \), \( x_2 = 3 \), \( y_2 = -2 \)

Step 3 - Substitute the Coordinates into the Slope Formula

Substitute the coordinates into the slope formula \[ m = \frac{-2 - 4}{3 - (-1)} \]. Simplify the equation step-by-step.

Step 4 - Calculate the Numerator

Calculate the numerator: \( -2 - 4 = -6 \).

Step 5 - Calculate the Denominator

Calculate the denominator: \( 3 - (-1) = 3 + 1 = 4 \).

Step 6 - Calculate the Slope

Combine the results from the numerator and the denominator: \[ m = \frac{-6}{4} = -1.5 \]

Step 7 - Round to the Nearest Hundredth

In this case, the value is already to the nearest hundredth, so no further rounding is required.

Key Concepts

slope formula

To find the slope of a line passing through two points, we use the slope formula. This formula helps us determine how steep the line is. The slope, often represented by the letter m, is calculated using the coordinates of the two points through which the line passes.
The formula is given by: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here, \(x_1\) and \(y_1\) are the coordinates of the first point, while \(x_2\) and \(y_2\) are the coordinates of the second point. By plugging in the values of these coordinates, we can calculate the slope.
Let's illustrate this with an example. Suppose we have points A (-1, 4) and B (3, -2). We identify the coordinates: \( x_1 = -1 \), \( y_1 = 4 \), \( x_2 = 3 \), and \( y_2 = -2 \). By substituting these values into the slope formula, we get:
\[ m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -1.5 \]
This means the slope of the line passing through points A and B is -1.5.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system employs coordinates to specify the positions of points on a plane. In a two-dimensional coordinate system, any point can be described using an ordered pair (x, y), where x represents the horizontal distance and y represents the vertical distance.
When dealing with lines in coordinate geometry, the slope is an important concept as it tells us how a line inclines or declines. For example:

• A positive slope means the line is rising as it moves from left to right.
• A negative slope means the line is falling as it moves from left to right.
• A slope of zero indicates a perfectly horizontal line.
• An undefined slope (division by zero) indicates a perfectly vertical line.

Understanding these aspects helps in graphing the lines and solving various problems related to the positions and shapes of geometric figures on a coordinate plane.

linear equations

Linear equations represent straight lines when graphed on a coordinate plane. They are typically written in the form: \[ y = mx + b \]
Here, m represents the slope of the line and b represents the y-intercept, which is the point where the line crosses the y-axis.
Every linear equation can be graphed using its slope and intercept. For instance, if we know the slope is -1.5 (as calculated previously) and assume an intercept b, we can write the linear equation for the line passing through points A (-1, 4) and B (3, -2) as:
\[ y = -1.5x + b \]
To find the exact value of b, we would substitute the coordinates of one of the points into the equation and solve for b. Let's use point A (-1, 4):

• Substitute \( x = -1 \) and \( y = 4 \): \[ 4 = -1.5(-1) + b \]
• Simplify to solve for b: \[ 4 = 1.5 + b \]
  b = 4 - 1.5 = 2.5

So, the equation of the line is: \[ y = -1.5x + 2.5 \]
Understanding how to derive and interpret linear equations is fundamental in coordinate geometry, as it forms the basis for more complex geometric concepts and calculations.