Problem 29 - 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. $$\left\\{\begin{array}{l}x-2 y=4 \\ 2 x-4 y=8\end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has infinite solutions, represented as {(x,y) | y = 0.5x - 2}.

Step 1: Convert the equations into slope-intercept form (y = mx + b)

Divide the second equation by 2 to form a new system: \[\begin{array}{l}x - 2y = 4 \ x - 2y = 4\end{array}\] In slope-intercept form, these become Y = 0.5X - 2.

Step 2: Graph the equations

These equations represent the same line when graphed, indicating they have infinite solutions.

Step 3: State the solutions in set notation

Since the lines coincide, there are an infinite number of points at which they intersect. Hence, the solution in set notation is {(x,y) | y = 0.5x - 2}.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is pivotal when it comes to graphing linear equations easily. The slope-intercept form is expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis.

The slope provides valuable information on the direction and steepness of the line. A positive slope means the line ascends from left to right, while a negative slope indicates it descends. The greater the magnitude of the slope, the steeper the line. In the provided exercise, both equations \( x - 2y = 4 \) and \( 2x - 4y = 8 \) were manipulated into the slope-intercept form, revealing that \( y = 0.5x - 2 \) for both. This form made it evident that the lines would be identical when graphed, as they share the same slope and y-intercept.

Set Notation

Set notation is a method of expressing a collection of objects, numbers, or, in the context of solving systems of equations, solutions. In mathematics, we use curly braces \( \{ \} \) to describe a set, and we can employ descriptors like brackets, vertical bars, and colon to articulate the conditions that members of the set must satisfy.

For instance, the solution set \( \{(x,y) | y = 0.5x - 2\} \) articulates that it consists of ordered pairs \( (x, y) \) where the second element is half of the first minus two. In the case of our original exercise, the use of set notation was instrumental in succinctly demonstrating the infinite number of solutions.

Infinite Solutions

When we talk about a system of equations having 'infinite solutions', it means that there isn't just one specific point of intersection between the lines- they overlap completely, sharing all points. In graphical terms, this translates to one line laying exactly on top of the other.

In our step-by-step solution, after converting both equations into the slope-intercept form and graphing them, it was clear that both equations represent the same line. This is a classic case of infinite solutions, as whatever point you choose on the line will satisfy both equations. Recognizing the possibility of infinite solutions helps students understand that not all systems of equations are solvable with distinct, singular solutions.