Problem 38 - Algebra and Trigonometry

Question

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 given system of equations represents the same line and thus has infinitely many solutions. The solution set, in set notation, is: \(\left\{ (x, 2x-1) \, | \, x \in \mathbb{R} \right\}\)

Step 1: Check if the Equations are Equivalent

Examine the given two equations \(4x - 2y = 2\) and \(2x - y = 1\). We can see that the second equation is exactly the half of the first equation, which implies the two equations represent the same line and thus have infinitely many solutions.

Step 2: Express the Solution in Set Notation

The set of all points on the line defined by either of the equations is the solution. The line is \(2x - y = 1\) or \(y = 2x - 1\). Therefore, the set of solutions is the set of all points \((x, y)\) on this line. The solution can be expressed in set notation as: \(\left\{ (x, 2x-1) \, | \, x \in \mathbb{R} \right\}\), where \(\mathbb{R}\) represents the set of all real numbers.

Key Concepts

Infinitely Many Solutions

In a system of equations, sometimes both equations describe the same line. When this happens, the system is said to have infinitely many solutions. This means that any point on the line is a solution to both equations. For the system in the exercise,

• The first equation is \(4x - 2y = 2\).
• The second equation is \(2x - y = 1\).

If you take the second equation and multiply every term by 2, you get the first equation. This confirms they are equivalent and describe the same line. So, every point on that line is a solution, giving us infinitely many solutions.

Set Notation

Set notation is a way of describing a collection of objects, known as a set. In mathematics, set notation is often used to express the solution set of an equation or system of equations.
For the given system, the solution is the set of all points \((x, y)\) that satisfy the equation \(y = 2x - 1\). This is written as:
\[ \{ (x, 2x-1) \, | \, x \in \mathbb{R} \} \]This notation tells us:

• The set consists of all pairs \((x, y)\),
• Each \(y\) is calculated as \(2x - 1\),
• \(x\) can be any real number \(\mathbb{R}\).

Thus, set notation provides a compact way to describe the infinite number of solutions that lie on the line.

Equivalent Equations

Equations are considered equivalent if they have the same solutions. In the context of a system of linear equations, if scaling or manipulating one equation results in the other, then they are equivalent.
In our exercise:

• Original equations are \(4x - 2y = 2\) and \(2x - y = 1\).
• By scaling the second equation by a factor of two, it matches the first one.

This manipulation confirms the two equations are equivalent, meaning they represent the same line, leading to infinitely many solutions.

Linear Equations

A linear equation in two variables, such as \(ax + by = c\), represents a straight line when plotted on a coordinate plane. Each point on this line is a solution to the equation.
The given equations are:

• \(4x - 2y = 2\)
• \(2x - y = 1\)

Both are linear equations, and simplifying one shows it is the same line as the other. A unique feature of linear equations in a system is that they can form parallel lines, intersect at a point, or, as in this exercise, overlap completely.
This overlap is what gives the system infinitely many solutions, as explained above.