Problem 36 - Introductory Algebra for College Students

Question

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}2 x-y=0 \\ y=2 x\end{array}\right.$$

Step-by-Step Solution

Answer

The solution for the system is \(\{(x,y)|x,y \in R\}\). This means the system has infinite solutions because both equations represent the same line in the graph.

Step 1: Rewrite the Equations

Rewrite the given equations in slope-intercept form \(y=mx+b\), where m is the slope, and b is the y-intercept. First equation becomes: \(y = 2x\). The second equation is already in the correct form: \(y=2x\).

Step 2: Graphing

Draw the graphs of the two equations on the same plane. Since both equations represent the same line \(y=2x\), there will only be one line on your graph.

Step 3: Find the Solution Set

Since both equations represent the same line, every point on the line is a valid solution for the system. The solution set would be all real numbers, expressed in set notation like this: \(\{(x,y)|x,y \in R\}\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential when dealing with linear equations. It is expressed as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis. This form allows you to quickly identify these two key characteristics of a line.

For the exercise under consideration, the goal is to rewrite the given equations into this form. The first equation, \(2x - y = 0\), rearranges to \(y = 2x\) upon adding \(y\) to both sides. Notice here, the slope \(m\) is 2, and the y-intercept \(b\) is 0, indicating that our line crosses the y-axis at the origin \(0,0\). The second equation is already provided in the required form.

What's important is that the slope-intercept format not only simplifies graphing but also makes comparing different lines straightforward. If two lines have the same slope and y-intercept, as with our problem, they are, in fact, the same line, which is a unique scenario in systems of equations.

Graphing Linear Equations

Graphing linear equations involves plotting the function of \(y\) against \(x\) on a two-dimensional plane to visualize the relationship between the variables.

Breakdown of the process starts with identifying the slope and y-intercept from the slope-intercept form, which dictates the starting point and the angle of the line. For the equation \(y = 2x\), we begin at the y-intercept \(0,0\), and for every unit increase in \(x\), \(y\) increases by 2 units due to the slope of 2. This slope tells us that the line rises steeply, ascending two units vertically for each single unit moved horizontally.

After plotting the intercept, you draw a line through this point that rises (positive slope) or falls (negative slope) at an angle consistent with the slope value. In scenarios where two equations share the same slope and intercept, they will overlay perfectly on one another on the graph, indicating that they are essentially one and the same line—a key observation for the given exercise.

Set Notation

Set notation is a method used to describe a collection of elements that satisfy certain conditions. In the context of systems of equations, set notation is employed to express the solution set concisely.

In our exercise, since both linear equations graph to the same line, there is an infinite number of solutions—every point on the line is a solution to the system. To express this infinite set of solutions, we use set notation: \(\{ (x, y) | x, y \in \mathbb{R} \}\), which reads as 'the set of all ordered pairs \(x\) and \(y\) such that \(x\) and \(y\) are elements of the real numbers.' This notation effectively communicates the idea of 'all possible points on the line' without needing to list each point individually.

Recognizing how to interpret and utilize set notation is invaluable for clearly conveying solutions to problems where individual enumeration of each solution is impractical or impossible, such as in the case of our linear equation with infinitely many solutions.