Problem 39 - Elementary Algebra

Question

Find the slope of the line passing through the points (4,-3) and (1,-6) .

Step-by-Step Solution

Answer

Answer: The slope of the line is 1.

Step 1: Identify the Coordinates

We have the points (4, -3) and (1, -6), so the coordinates are as follows: x1 = 4 y1 = -3 x2 = 1 y2 = -6

Step 2: Use the Slope Formula

Plug the coordinates into the slope formula: m = (y2 - y1) / (x2 - x1)

Step 3: Substitute the Coordinates

Substitute the coordinates we found in Step 1 into the formula: m = (-6 - (-3)) / (1 - 4)

Step 4: Simplify the Expression

Simplify the expression further to find the slope: m = (-6 + 3) / (-3) m = (-3) / (-3)

Step 5: Calculate the Slope

Finally, calculate the slope: m = 1 Thus, the slope of the line passing through the points (4, -3) and (1, -6) is 1.

Key Concepts

coordinate geometry

Coordinate geometry, sometimes known as analytic geometry, is a mathematical concept that connects algebra and geometry. It allows us to visually represent algebraic equations in a two-dimensional plane. The coordinate plane consists of two perpendicular lines, known as the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants, creating a grid where each point is represented by an ordered pair \(x, y\).
This framework makes it easier to understand geometric concepts such as lines, curves, and angles by expressing them in numerical form. For example, a line in the coordinate plane can be represented by an equation and examined using the coordinates of points it passes through. Understanding coordinate geometry is essential for solving problems involving distance, midpoint, and slope, as it creates a bridge between numerical and spatial reasoning. By using coordinates, it becomes easier to apply algebraic methods to determine the properties of geometric figures.

slope formula

The slope formula is a fundamental principle in coordinate geometry for describing the steepness or incline of a line. It is often symbolized by the letter \(m\) and is calculated using the coordinates of two distinct points on the line. If you have points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \(m\) is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• \(y_2 - y_1\) represents the change in the y-coordinates (vertical change).
• \(x_2 - x_1\) represents the change in the x-coordinates (horizontal change).

The slope is essentially a ratio that describes how much the \(y\)-value of a line changes for every unit of change in the \(x\)-value. A positive slope indicates a line that rises as it moves from left to right, while a negative slope indicates a line that falls. If the slope is zero, the line is horizontal, reflecting no vertical change, and if undefined (when the denominator is zero), the line is vertical. Understanding the slope helps in analyzing and graphing linear relationships effectively.

linear equations

Linear equations represent straight lines in coordinate geometry and are typically expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept—the point where the line crosses the y-axis.
Such equations model relationships where the change between variables is constant. This constant change makes linear equations incredibly useful for predicting and analyzing trends in data sets.
To graph a linear equation, you need:

• The slope \(m\) to determine the line’s steepness.
• The y-intercept \(b\) which gives the starting point on the y-axis.

By plugging in values for \(x\), you can solve for \(y\) and thus explore the various points along the line. Linear equations not only help in graphing but also in solving practical problems such as calculating budgets, profits, or anything that changes at a steady rate. Understanding and being able to manipulate linear equations is a valuable skill in both mathematics and everyday reasoning.