Problem 24 - Algebra 1

Question

Plot the points and find the slope of the line passing through the points. $$(0,6),(8,0)$$

Step-by-Step Solution

Answer

The slope of the line passing through points (0,6) and (8,0) is -0.75

Step 1: Identifying the Coordinates

The first point is (0,6) where 0 is x1 and 6 is y1. The second point is (8,0) where 8 is x2 and 0 is y2. So, the coordinates are as follows: x1=0, y1=6, x2=8, y2=0.

Step 2: Inputting the Coordinates into the Slope Formula

Substitute the coordinates into the slope formula: \(m = \frac{(y2 - y1)}{(x2 - x1)} = \frac{(0 - 6)}{(8 - 0)}\)

Step 3: Calculating the Slope

Simplify the expression: \(m = \frac{-6}{8} = -0.75\)

Step 4: Plotting The Points

Plot the two points (0,6) and (8,0) on a graph. Then draw the line through these points. The line will slope downwards from left to right, reflecting the negative slope calculated.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometric figures through a coordinate system. This system uses numerical values, or coordinates, to represent points on a two-dimensional plane.

In the context of finding the slope of a line, the coordinate system we use is typically the Cartesian plane. This plane has two axes: the horizontal x-axis and the vertical y-axis. To locate a point on this plane, we use an ordered pair of numbers (x, y). The first number, x, indicates the position relative to the x-axis, while y denotes the position relative to the y-axis.

Using coordinate geometry, we can plot points, draw geometric figures, and calculate distances, midpoints, and slopes. It's a fundamental tool in the fields of mathematics, engineering, and physics, as it allows the visualization and analysis of geometric problems in a numeric way.

Slope Formula

The slope of a line is a measure of its steepness, which can be positive, negative, zero, or undefined. In the context of a line on a coordinate plane, we often refer to the slope as 'rise over run'. The slope formula is expressed as: \[m = \frac{(y2 - y1)}{(x2 - x1)}\]

Here, m represents the slope, and (x1, y1) and (x2, y2) are the coordinates of any two points on the line. If the slope is positive, the line ascends from left to right; if negative, it descends. A zero slope means the line is horizontal, and an undefined slope corresponds to a vertical line.

When using the formula, subtract the y-coordinate of the first point from the y-coordinate of the second, and do the same for the x-coordinates. Simplifying the fraction will give the slope of the line.

Plotting Points on a Graph

Plotting points on a graph involves placing dots at the coordinates that correspond to the values given for each point. To locate where to plot a point, start from the origin, which is where the x-axis and y-axis intersect, at the coordinates (0,0). Move horizontally to the right or left based on the x-coordinate's value and vertically up or down based on the y-coordinate.

When plotting the points for a line, such as \[(0,6)\] and \[(8,0)\], you would place a point at 6 units above the origin for (0,6) and 8 units to the right of the origin for (8,0). After plotting these two points, a line is drawn through them to visualize the slope. In this case, because the points create a straight line that slants downward from left to right, the slope is negative, indicating that for every step we move to the right along the x-axis, the line falls a certain amount downwards.