Problem 31 - Elementary Algebra

Question

You are given one point on a line and the slope of the line. Find the coordinates of three other points on the line. $$(4,-5), m=-2$$

Step-by-Step Solution

Answer

The three other points are (5, -7), (6, -9), and (7, -11).

Step 1: Understand the problem

We are given a point on a line, (4, -5), and the slope of the line, \( m = -2 \). Our task is to find the coordinates of three other points that lie on the same line.

Step 2: Slope definition

Recall that the slope \( m \) is defined as the change in \( y \) divided by the change in \( x \). This means that if we move from one point to another on the line, \( \Delta y = m \times \Delta x \). Here \( m = -2 \).

Step 3: Find the second point

Start from the given point (4, -5) and choose a \( \Delta x \). Let's choose \( \Delta x = 1 \). Then \( \Delta y = -2 \times 1 = -2 \). The new \( y \)-coordinate is \( -5 + \Delta y = -5 - 2 = -7 \). So the second point is (5, -7).

Step 4: Find the third point

Starting again from (5, -7), choose \( \Delta x = 1 \). Then \( \Delta y = -2 \times 1 = -2 \). The new \( y \)-coordinate is \( -7 - 2 = -9 \). So the third point is (6, -9).

Step 5: Find the fourth point

From the third point (6, -9), again choose \( \Delta x = 1 \). Then \( \Delta y = -2 \times 1 = -2 \). The new \( y \)-coordinate is \( -9 - 2 = -11 \). So the fourth point is (7, -11).

Key Concepts

Understanding Linear Equations

Linear equations are fundamental to algebra and represent straight lines when graphed. They are typically expressed in the slope-intercept form, which is \(y = mx + b\), where

• \(m\) is the slope of the line, defining its steepness.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

In this exercise, the slope \(m\) is given as \(-2\), meaning for every one unit increase in \(x\), \(y\) decreases by two units. Linear equations help in predicting and understanding various phenomena by showing a constant rate of change. The relationship between \(x\) and \(y\) in a linear equation is significantly easy to manage mathematically, making them a cornerstone in problem-solving.

Exploring Coordinate Geometry

Coordinate geometry, or analytical geometry, is the study of geometry using a coordinate system such as the Cartesian plane. It enables us to determine the position of points and understand shapes and sizes in two dimensions.
In a Cartesian plane, each point is identified by an \((x, y)\) coordinate. The line's position and direction are defined by its slope and a point on the line. Given in the exercise, the point \((4, -5)\) and the slope \(m = -2\). This setup allows us to use the slope to find other points on the line by moving horizontally (changing \(x\)) and adjusting vertically (changing \(y\) according to the slope).
This computation exposes the relationship between algebraic equations and their graphical representation, solidifying understanding of abstract mathematical concepts through visual interpretation. The power of coordinate geometry lies in its ability to convert numerical data to graphical, making interpretations and connections between different data points easier.

Algebra Problem-Solving Techniques

In algebra, problem-solving often involves translating real-world scenarios into mathematical expressions. To solve this exercise, start from a known point and apply the slope concept. Here's the approach:
1. **Start with the known point**: \((4, -5)\).

• The slope \(m = -2\) suggests every unit increase in \(x\) results in a \(-2\) change in \(y\).
• To find another point, increment \(x\) by 1 twice more, each time subtracting two from the previous \(y\) value to maintain the slope. This gives the new point \((5, -7)\), then \((6, -9)\), and finally \((7, -11)\).

2. **Consistent Patterns**: With an understanding of the slope, you can effectively establish a line's points through basic algebraic manipulation. Such exercises sharpen skills in employing consistent mathematical patterns and operations to solve equations.
3. **Real-World Relevance**: Algebraic concepts like slope are practical; think of slopes in roads or data trends. Mastering these problem-solving techniques enriches students with tools applicable in various disciplines, such as physics or economics.