Problem 56 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. $$ (-5,4),(-1,0) $$

Step-by-Step Solution

Answer

Answer: The slope of the line passing through the points (-5, 4) and (-1, 0) is -1.

Step 1: Identify the coordinates of the given points

We are given these two points: \((-5, 4)\) and \((-1, 0)\). We can identify \(x_1 = -5\), \(y_1 = 4\), \(x_2 = -1\), and \(y_2 = 0\).

Step 2: Apply the slope formula

Recall that the slope formula is given by: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Plug the coordinates we identified in step 1 into the formula: $$ m = \frac{0 - 4}{-1 - (-5)} $$

Step 3: Simplify the expression

Now, simplify the fraction in the equation to find the slope: $$ m = \frac{-4}{4} $$ $$ m = -1 $$

Step 4: State the final answer

The slope of the line passing through the points \((-5, 4)\) and \((-1, 0)\) is \(m = -1\).

Key Concepts

Coordinate Geometry

Coordinate geometry, a fundamental aspect of geometry, helps us understand relations and distances between points on a plane. It does this by using a coordinate system to represent geometric shapes in a numerical manner. In the cartesian coordinate system, every point on the plane is described by an ordered pair \(x, y\), representing horizontal and vertical positions respectively.

• **Points**: These are represented by their coordinates, like \( (-5, 4) \) and \( (-1, 0) \).
• **Lines**: A line is entirely described by its slope and a point on it or two distinct points through which it runs.

This method bridges algebra and geometry, allowing equations and graphs to blend and give a clearer picture of spatial relationships and structures.

Linear Equations

Linear equations are algebraic expressions that yield a straight line when graphed on a coordinate plane. The most common form is the slope-intercept form, given as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• **Slope**: It illustrates the direction and steepness of a line, calculated using the difference in y-coordinates divided by the difference in x-coordinates between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• **Relationship**: Linear equations are essential in defining linear relationships and help in predicting and interpreting data trends.

Linear equations form the backbone of various mathematical concepts and applications, setting a solid foundation for more complex calculations.

Points on a Plane

Understanding points on a plane is crucial, as they form the base of coordinate geometry. Each point is defined by a set of two numbers: the x-coordinate and the y-coordinate. These coordinates let you pinpoint any location precisely on the cartesian plane.

• **X-Coordinate**: Tells how far along the horizontal axis a point is.
• **Y-Coordinate**: Indicates the vertical distance from the origin.
• **Plotting**: To graph a point, start at the origin (0,0), move x units parallel to the x-axis, then y units parallel to the y-axis.

This system facilitates the layout of mathematical problems, making it easier to visualize equations and their solutions. Recognizing and correctly using points on a plane lays the groundwork for understanding more advanced geometric and algebraic concepts.