Problem 38 - College Algebra Essentials

Question

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{l} 4 x-2 y=2 \\ 2 x-y=1 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions. The solution set is \(\{(x, 2x - 1) | x \in \mathbb{R}\}\)

Step 1: Write the System of Equations

Begin by writing down the given system of linear equations as:\[ \begin{{array}}{{l}} 4x - 2y = 2 \ 2x - y = 1 \end{{array}}\]

Step 2: Simplify the Equations

Notice that the first equation can be simplified by dividing by 2 on both sides. It becomes \(2x - y = 1\), which is the same as the second equation. So, we have a simplified system of equations:\[ \begin{{array}}{{l}} 2x - y = 1 \ 2x - y = 1 \end{{array}}\]

Step 3: Solve for the Variables

Here, both equations are exactly the same. That means every solution (x, y) to the first equation will also be a solution to the second equation. Thus, we say that the system of equations has infinitely many solutions.

Step 4: Write the Solution Set

The solution set is all (x, y) that satisfy the equation \(2x - y = 1\). To express this in set notation, we solve for y to get \(y = 2x - 1\). The solution set is then expressed as:\[\{(x, 2x - 1) | x \in \mathbb{R}\}\]where \(x \in \mathbb{R}\) means x is any real number.

Key Concepts

Solution Sets

In mathematics, a solution set is a collection of all possible solutions to a system of equations. When we talk about a system like the one in your exercise, we are looking for pairs of numbers

• One number for each variable that make all the equations in the system true at the same time.

For the equations given in the exercise: \[ \begin{array}{l} 4x - 2y = 2 \ 2x - y = 1 \end{array} \]
we found that after simplifying them, both equations are essentially the same.
This means that any (x, y) pair that solves one equation will solve the other as well.
Thus, the solution set for this system isn't just a single solution, but rather infinite pairs of solutions.
We can write down all these pairs in a concise manner using set notation.

Infinitely Many Solutions

Sometimes, a system of equations doesn't have just one solution, but rather infinitely many. This happens when the equations are essentially the same, just rewritten in different forms.
In simpler terms, every x-value paired with a corresponding y-value that solves one equation will also solve the other.
Let's take the simplified system you found: \[ \begin{array}{l} 2x - y = 1 \ 2x - y = 1 \end{array} \]
Here, both equations are identical, meaning they represent the same line in the plane.
Because they overlap completely, any point on this line is a solution.
Thus, there are infinitely many solution pairs \((x, y)\) for this system.
We can describe this set of solutions using set notation to keep it organized and clear.

Set Notation

Set notation is a mathematical shorthand used to list all the solutions of a system in a neat format. When you have infinitely many solutions, as we do in the exercise, set notation is especially useful.
For the system: \[ \begin{array}{l} 2x - y = 1 \ 2x - y = 1 \end{array} \]
After solving for \(y\), we get \(y = 2x - 1\). This equation describes a line, and every point on this line is a solution to our system.
The solution set in set notation is thus written as:

• \(\{(x, 2x - 1) \mid x \in \mathbb{R}\}\)

This expression means "all pairs \((x, 2x - 1)\)" where \(x\) is any real number \(\mathbb{R}\).
"\(|\)" is read as "such that," indicating the relation between \(x\) and \(y\).
Set notation offers an efficient way to describe solution sets that contain an infinite number of elements.