Problem 49 - Algebra 1

Question

Use the concept of slope to decide whether the points \((-2,4),(2,-2),\) and \((6,0)\) are on the same line. Explain your reasoning and include a diagram.

Step-by-Step Solution

Answer

No, the points (-2, 4), (2, -2) and (6, 0) are not on the same line. This is because the slope between the first two points is not equal to the slope between the last two points.

Step 1: Calculating the slope between point 1 and point 2

Using the slope formula \((y2 - y1) / (x2 - x1)\), where point 1 is (-2,4) and point 2 is (2,-2), the slope between these two points would be \((-2 - 4) / (2 - (-2)) = -6 / 4 = -1.5\)

Step 2: Calculating the slope between point 2 and point 3

Now calculate the slope between point 2 (2,-2) and point 3 (6,0). Following the same principle, we find that the slope is \((0 - (-2)) / (6 - 2) = 2 / 4 = 0.5\).

Step 3: Comparing the slopes

Now we notice that the slopes found are not equal. The slope between point 1 and point 2 is -1.5, while the slope between point 2 and point 3 is 0.5. Since the slopes are not the same, the points are not on the same line.

Key Concepts

Linear Equations

A linear equation is a fundamental concept in algebra that represents a straight line when graphed. These equations have the general form of \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

• The slope \( m \) indicates the steepness of the line and its direction (rising, falling, or constant).
• The y-intercept \( b \) is the point where the line crosses the y-axis.

In the context of our exercise, determining if three points are on the same line involves checking if all sections of the potential line have the same slope. Thus, the consistency of the slope across all pairs of points tells us if they form the same line—the essence of linear equations in coordinate geometry. Understanding these properties helps solve many real-life problems, such as predicting trends or making forecasts using linear data.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe points, lines, and shapes using coordinates. This system allows us to use equations to represent geometric figures, making complex concepts more tangible.To understand if three points

• \(( -2, 4 )\)
• \(( 2, -2 )\)
• \(( 6, 0 )\)

lie on the same line, we calculate the slopes between pairs of points. If these slopes are consistent across the board, we know the points lie on the same line, as a single straight line should have the same slope between any two points.This logical assessment of slopes forms the crux of coordinate geometry, helping us visualize and evaluate the relationships between different points on a plane through calculations and equation plotting.

Point-Slope Formula

The point-slope formula is a methodical way of finding a linear equation when you know one point on the line and the line's slope. The formula is expressed as: \[ y - y_1 = m(x - x_1) \]In this formula:

• \((x_1, y_1)\) represents a known point on the line.
• \(m\) is the slope of the line.

the point-slope form can be converted to the slope-intercept form \( y = mx + b \), providing a versatile way to express a linear equation.When applied to the exercise, once the slope was calculated between two points, the point-slope formula could have been used to express the line's equation. Verifying if another point fits within this equation would confirm if it lies on the same line. While not used explicitly in the solution, understanding the point-slope formula enriches our ability to approach problems involving lines and slopes with multiple strategies.