Question

Find the solution to the given linear system. If the system has infinite solutions, give 2 particular solutions. $$ \begin{aligned} -2 x_{1}+4 x_{2}+4 x_{3} &=6 \\ x_{1}-3 x_{2}+2 x_{3} &=1 \end{aligned} $$

Short answer

The system has infinite solutions. Two particular solutions are: \((-11, -4, 0)\) and \((-1, 0, 1)\).

Step-by-step solution

Identify the System of Equations

We are given a system of two linear equations with three variables. The system is:\[\begin{aligned}-2x_1 + 4x_2 + 4x_3 &= 6 \x_1 - 3x_2 + 2x_3 &= 1\end{aligned}\]

Express One Variable in Terms of Others

Let's solve the second equation for \(x_1\):\[x_1 = 3x_2 - 2x_3 + 1\]

Substitute into the First Equation

Substitute \(x_1 = 3x_2 - 2x_3 + 1\) into the first equation:\[-2(3x_2 - 2x_3 + 1) + 4x_2 + 4x_3 = 6\]Simplify the equation:\[-6x_2 + 4x_3 - 2 + 4x_2 + 4x_3 = 6\]\[-2x_2 + 8x_3 = 8\]\[-x_2 + 4x_3 = 4\]

Solve for One Variable

Solve for \(x_2\) in terms of \(x_3\):\[-x_2 + 4x_3 = 4\]\[x_2 = 4x_3 - 4\]

Express All Variables in Terms of Parameters

At this point, express all variables in terms of a parameter, say \(t\), for \(x_3\):\[\begin{aligned}x_3 &= t \x_2 &= 4t - 4 \x_1 &= 3(4t - 4) - 2t + 1 = 12t - 12 - 2t + 1 = 10t - 11\end{aligned}\]

Identify Two Particular Solutions

Substitute specific values for \(t\) to get particular solutions. For example, let:- \(t = 0\): \[ \begin{aligned} x_3 &= 0 \ x_2 &= 0 - 4 = -4 \ x_1 &= 10(0) - 11 = -11 \end{aligned} \] So, one particular solution is \((-11, -4, 0)\).- \(t = 1\): \[ \begin{aligned} x_3 &= 1 \ x_2 &= 4(1) - 4 = 0 \ x_1 &= 10(1) - 11 = -1 \end{aligned} \] So, another particular solution is \((-1, 0, 1)\).

Key concepts

Infinite Solutions

In some systems of linear equations, you may find that there are infinite solutions. This happens when there are more variables than equations, leading to an underdetermined system.
For the given system of equations: \[-2x_1 + 4x_2 + 4x_3 = 6\] \[x_1 - 3x_2 + 2x_3 = 1\] we have three variables (\[x_1, x_2,\] and \[x_3\]) but only two equations.
This means the system is unable to pinpoint a single unique solution, and instead, an entire set of solutions exists that satisfy all equations.
The key here is expressing all variables in terms of one free variable, which represents an entire line or plane of solutions.
Thus, the system is said to have infinite solutions, allowing flexibility in choosing particular values.

System of Equations

A system of equations consists of two or more equations that share common variables. When you solve a system, you're trying to find values for these variables that make all the equations true at the same time.
In the system provided, we have: \[-2x_1 + 4x_2 + 4x_3 = 6\] \[x_1 - 3x_2 + 2x_3 = 1\] Systems can be categorized by the number of solutions:

• No solution: The equations represent parallel lines that never intersect.
• One solution: The equations intersect at exactly one point.
• Infinite solutions: The equations represent the same line or plane and overlap completely.

For this exercise, our system has more variables than equations, leading primarily to either no or infinite solutions. Using algebraic methods like substitution or elimination, we attempt to solve such systems.

Particular Solutions

When a system of equations has infinite solutions, we can find specific solutions called particular solutions. These are obtained by assigning specific values to the free variables.
For the system given, we expressed all variables in terms of a parameter \[t\].
For example:

• If \[t = 0\], the solution is \[(-11, -4, 0)\].
• If \[t = 1\], the solution is \[(-1, 0, 1)\].

These two different sets of \((x_1, x_2, x_3)\) values satisfy both original equations and exemplify the numerous solutions that exist within the infinite solution set.
Choosing particular values for \[t\] helps illustrate the behavior of the system and its solutions in a more tangible way.