Question

Find the solution to the given linear system. If the system has infinite solutions, give 2 particular solutions. $$ \begin{array}{l} -x_{1}+5 x_{2}=3 \\ 2 x_{1}-10 x_{2}=-6 \end{array} $$

Short answer

The system has infinite solutions; two particular solutions are \((-3, 0)\) and \((2, 1)\).

Step-by-step solution

Write the System of Equations

We are given the following system of equations: 1. \(-x_1 + 5x_2 = 3\)2. \(2x_1 - 10x_2 = -6\).Our goal is to solve this system for \(x_1\) and \(x_2\).

Simplify and Compare the Equations

Multiply the first equation by 2 to make the coefficient of \(x_1\) the same:\[-2x_1 + 10x_2 = 6\].Notice that the second equation is \[2x_1 - 10x_2 = -6\].By adding these, check if the equations are dependent.

Add the Modified Equations

Adding \(-2x_1 + 10x_2 = 6\) and \(2x_1 - 10x_2 = -6\), we get:\[0 = 0\].This indicates the system of equations is dependent, meaning they represent the same line; therefore, the system has infinitely many solutions.

Express One Variable in Terms of the Other

We will express one variable in terms of the other using one of the equations. Let's use the first equation:\(-x_1 + 5x_2 = 3\). Solving for \(x_1\), we have:\[x_1 = 5x_2 - 3\].

Find Particular Solutions

Using the equation \(x_1 = 5x_2 - 3\), choose arbitrary values for \(x_2\) to find particular solutions:- If \(x_2 = 0\), then \(x_1 = 5(0) - 3 = -3\). Hence, one solution is \((x_1, x_2) = (-3, 0)\).- If \(x_2 = 1\), then \(x_1 = 5(1) - 3 = 2\). Hence, another solution is \((x_1, x_2) = (2, 1)\).

Key concepts

Infinite Solutions

When we talk about a system of linear equations having infinite solutions, we refer to the situation where the equations describe the same line on a graph. In such a case, any point on this line represents a solution to the system. This occurs when the equations are consistent and dependent. In simpler terms, there's no unique solution because there are countless solutions that satisfy both equations simultaneously.
To identify infinite solutions, look for equations that can be manipulated to appear as identical. For example, multiplying or dividing one equation to match another is a clear indicator of dependency, leading to infinite solutions.

Dependent Equations

Dependent equations occur when one equation is a multiple of another. This means they graph the same line, and thus, are effectively "the same" equation. When linear equations are dependent, they are not independent of each other.
In the context of our task, after multiplying the first equation by 2, it becomes obvious that both equations express the same relationship between variables. So if you add the two equations together, you'll get a true statement like 0 = 0, confirming their dependence.
When solving linear systems, checking for dependency can save time and clarify that no unique solution exists.

Particular Solutions

Particular solutions are specific solutions that satisfy the entire system of equations. When a system has infinite solutions, you can select arbitrary values for one variable, then solve to find the corresponding value of the other variable.
This is helpful when you have a dependent system, as you can demonstrate solutions by picking straightforward values for one variable. In our example, choosing values like 0 or 1 for one variable leads to easy calculations to find particular solutions such as

• (-3, 0)
• (2, 1)

These are just two examples, and since there are infinite possibilities, you can choose many others.

Expressing Variables in Terms of Others

This technique involves rearranging one equation to express one variable in relation to another variable. It simplifies finding solutions for systems with infinite solutions because you can easily compute many solutions by plugging any value into one variable.
To apply this method, you typically solve one of the simplified equations for one variable. In this example, solving for \[x_1\], we have \[x_1 = 5x_2 - 3\].
Now, by substituting different values for \(x_2\), you immediately find corresponding \(x_1\) values, giving flexible solutions. This not only gives insights into the relationship between variables but also simplifies generating solutions as needed.