Problem 91 - Algebra and Trigonometry

Question

Exercises \(90-92\) will help you prepare for the material covered in the first section of the next chapter. Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect?

Step-by-Step Solution

Answer

The graphs of the equations \(x + 2y = 2\) and \(x - 2y = 6\) intersect at the point (2, -1).

Step 1: Rewrite the equations in slope intercept form

To graph linear equations, it's often easier to express them in the form \(y = mx + c\) where \(m\) is the slope and \(c\) is the y-intercept. The given equations \(x + 2y = 2\) and \(x - 2y = 6\) can be rewritten as \(y = -0.5x + 1\) and \(y = 0.5x - 3\) respectively.

Step 2: Graph the equations

Plot the two equations on the same rectangular coordinate system. Each equation forms a straight line. The y-intercept is the point at which the line crosses the y-axis, and the slope dictates the angle at which the line ascends or descends.

Step 3: Find the intersection point

The point where the two lines intersect on the graph represents the solution. In other words, it's the point that makes both equations true. To find this point algebraically, set the two equations equal to each other, solve for \(x\), and then substitute \(x\) into first equation to find corresponding \(y\) value. From \( -0.5x + 1 = 0.5x - 3\), we find \(x = 2\) and, substituting \(x = 2\) into \(y = -0.5x + 1\), we find \(y = -0.5 * 2 + 1 = -1\).

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is a streamlined way to write a linear equation. It follows the format \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

In simple terms, the slope \(m\) can be thought of as the 'steepness' or 'inclination' of the line, often calculated as 'rise over run' when looking at a graph. The y-intercept \(b\), on the other hand, provides a starting point for the line on the graph. Understanding how these components influence the line's appearance on a graph is crucial when graphing linear equations.

When we rewrite the equations \(x+2y=2\) and \(x-2y=6\) as \(y = -0.5x + 1\) and \(y = 0.5x - 3\), respectively, we make it easier to visualize and draw them on a rectangular coordinate system by using their slope and y-intercept. This not only simplifies the process but also aids in predicting the behavior of the line without plotting multiple points.

Navigating the Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane made up of two perpendicular axes: the horizontal x-axis and the vertical y-axis. This system is a foundational tool in graphing linear equations and analyzing their behavior.

The intersection of the two axes represents the origin, labeled as (0,0), serving as a central point from which all other coordinates are measured. Each point on this plane is represented by a pair of coordinates \( (x, y) \) that indicate its distance from the origin along the x-axis and y-axis.

To graph a linear equation on this system, you start by plotting the y-intercept on the y-axis. Then, you use the slope to determine the direction and steepness of the line, counting out the 'rise' over the 'run' from the y-intercept to find another point on the line. Plot this point and draw a straight line through both points to represent the equation visually. By plotting the equations given in the exercise, we create two distinct lines that will intersect at a specific point within this coordinate system.

Finding the Intersection Point of Lines

When graphing two or more linear equations on the same coordinate system, the intersection point is where the lines cross each other. This intersection represents a set of x and y values that satisfy both equations simultaneously, making it the solution to the system of equations.

To find this point algebraically, we equate the two equations and solve for one variable, typically x. Once x is found, we substitute it back into either of the original equations to find the corresponding y value. In the given exercise, we demonstrate this by setting \( -0.5x + 1 = 0.5x - 3 \) and solving for x to obtain \( x = 2 \). Substituting this x value into the equation \( y = -0.5x + 1 \) yields \( y = -1 \), giving us the intersection point (2, -1).

This intersection point is often an important solution in many fields, including mathematics, physics, and economics, as it represents the point at which two variables meet under two different conditions, providing valuable insights into their relationship.