Question

Find the solution to the given linear system. If the system has infinite solutions, give 2 particular solutions. $$ \begin{array}{l} 2 x_{1}+4 x_{2}=2 \\ x_{1}+2 x_{2}=1 \end{array} $$

Short answer

The system has infinite solutions; two particular solutions are (1, 0) and (-1, 1).

Step-by-step solution

Analyze the System

We have two equations given by \(2x_1 + 4x_2 = 2\) and \(x_1 + 2x_2 = 1\). Observe that the second equation is exactly half of the first one. This means both equations are equivalent, suggesting infinitely many solutions.

Reduce the System

Switch to a simpler form of the system by using just the second equation: \(x_1 + 2x_2 = 1\), which will help us find the solutions.

Express Variables

We can express one variable in terms of the other. From the equation \(x_1 + 2x_2 = 1\), solving for \(x_1\) gives: \(x_1 = 1 - 2x_2\).

Particular Solutions - Choose Values

Choose particular values for \(x_2\) to find specific solutions. Results should satisfy \(x_1 = 1 - 2x_2\).

Calculate Particular Solution 1

Choose \(x_2 = 0\) and substitute into \(x_1 = 1 - 2x_2\), so \(x_1 = 1\). The first particular solution is \((x_1, x_2) = (1, 0)\).

Calculate Particular Solution 2

Choose \(x_2 = 1\) and substitute into \(x_1 = 1 - 2x_2\), so \(x_1 = -1\). The second particular solution is \((x_1, x_2) = (-1, 1)\).

Key concepts

Infinite Solutions

In mathematics, a linear system is a collection of linear equations involving the same set of variables. A system can have three types of solutions: a unique solution, no solution, or infinitely many solutions. When we talk about infinite solutions, we mean there is not just one set of values for the variables that satisfy all the equations, but rather an entire family of solutions.
In the case of our exercise, the two given equations are not independent because one is a multiple of the other. They describe the same line in the coordinate plane. This means any point on this line is a solution to the system. Thus, there are infinitely many solutions. Such a scenario occurs often when equations in a linear system are equivalent.

• Infinite solutions occur if the equations are dependent or equivalent.
• A graph of the equations will show them overlapping.

Equivalent Equations

Equivalent equations are equations that represent the same relationship between variables, even though they may appear different at first glance. Two equations are equivalent if they have the same solution set. This can be spotted when one equation can be derived from the other through algebraic manipulation, such as multiplying or dividing through by a constant.
In our example, multiplying the second equation by 2 gives us the first equation:

• The first equation: \(2x_1 + 4x_2 = 2\).
• Twice the second equation: \(2(x_1 + 2x_2) = 2 \Rightarrow 2x_1 + 4x_2 = 2\)

This shows they are equivalent, indicating the system has infinite solutions. It's a critical concept when solving linear systems as it helps identify consistency in the system and identify redundant equations.

Particular Solutions

When a system of linear equations has infinitely many solutions, any specific set of values for the variables that satisfy all the equations can be called a particular solution. Even though there are countless solutions, sometimes specific solutions are required for practical purposes or deeper understanding.To find particular solutions, you pick arbitrary values for one variable and solve for others. In our problem, we use:

• For \(x_2 = 0\), we found \(x_1 = 1-2(0) = 1\), giving particular solution \((x_1, x_2) = (1, 0)\).
• For \(x_2 = 1\), we found \(x_1 = 1-2(1) = -1\), giving particular solution \((x_1, x_2) = (-1, 1)\).

Selecting specific solutions often gives a better intuition of the problem while still maintaining the generality of infinite solutions.

Variable Substitution

Variable substitution is a powerful method to simplify systems of equations and find solutions. The idea is to express one variable in terms of others using one of the equations, and then substitute this expression into the other equations. This reduces the number of variables and simplifies the system.
For our exercise, the equation \(x_1 + 2x_2 = 1\) allows us to substitute:

• Solve for \(x_1\) to get: \(x_1 = 1 - 2x_2\).

Now, \(x_1\) is expressed in terms of \(x_2\). This step is crucial because it allows us to easily generate infinite solutions by just deciding on different values for \(x_2\) and calculating \(x_1\) accordingly. It’s a systematic way to explore the solution space of the system, especially when dealing with infinite solutions.