Problem 19 - Algebra and Trigonometry

Question

\(17-36\) . Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x+y+z &=4 \\ x+3 y+3 z &=10 \\ 2 x+y-z &=3 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The linear system is consistent, and the solution is given by \((x, y, z) = (2t - 1, 3 - t, t)\), where \(t\) is any real number.

Step 1: Write the System of Equations

The given linear system of equations is: \( \begin{align*}x + y + z &= 4 \x + 3y + 3z &= 10 \2x + y - z &= 3\end{align*} \). These are three equations with three variables \(x\), \(y\), and \(z\).

Step 2: Eliminate a Variable

To simplify, let's eliminate \(x\) by subtracting the first equation from the second. This yields: \( (x + 3y + 3z) - (x + y + z) = 10 - 4 \), which simplifies to \( 2y + 2z = 6 \) or \( y + z = 3 \).

Step 3: Use Another Pair to Eliminate the Same Variable

Now, subtract the first equation from the third equation to further eliminate \(x\). That gives: \( (2x + y - z) - (x + y + z) = 3 - 4 \), simplifying to \( x - 2z = -1 \).

Step 4: Solve the Reduced System of Equations

Now, solve the two new equations: \( y + z = 3 \) and \( x - 2z = -1 \). From the first of these, express \(y\) as \( y = 3 - z \). Substitute \(y = 3 - z\) into an equation involving \(x\): \( x - 2z = -1 \).

Step 5: Solve for One Variable

Substitute \( z = 3 - y \) into \( x - 2z = -1 \), yielding \( x - 2(3 - y) = -1 \). Simplify to find \( x = -1 + 6 - 2y \), which reduces to \( x = 5 - 2y \).

Step 6: Solution in Parametric Form

Express \(x\), \(y\), and \(z\) in terms of a parameter, say \(t\). Let \(z = t\), then \( y = 3 - t \) and \( x = 5 - 2(3 - t) \). Simplify for \(x\) to get \( x = 5 - 6 + 2t = 2t - 1 \). Thus, the solution is: \( (x, y, z) = (2t - 1, 3 - t, t) \) for any real number \(t\).

Step 7: Check for Consistency

Substitute back to check against all original equations. \(x = 2t - 1\), \(y = 3 - t\), \(z = t\) satisfies all equations as verified by substitution. Therefore, the system is consistent.

Key Concepts

Parametric Form

In linear algebra, when dealing with systems of equations, finding a solution can lead us to what is known as a parametric form. This is especially useful when the system has infinitely many solutions, which means there are multiple values that satisfy all equations simultaneously.
For the given system of equations:

• \(x + y + z = 4\)
• \(x + 3y + 3z = 10\)
• \(2x + y - z = 3\)

Through various steps, we express each variable using another variable, often called a parameter, such as \(t\). Here, we discovered that:

• \(z = t\)
• \(y = 3 - t\)
• \(x = 2t - 1\)

This representation shows the general solution, giving us the flexibility to use any real number as \(t\) to obtain specific solutions. The parametric form is a compact way to describe potentially infinite solutions in a neat, understandable manner.

Variable Elimination

When solving a system of equations, one of the standard methods is variable elimination. This technique aims to reduce the number of variables until you can easily solve the system. It's achieved by adding, subtracting, or otherwise manipulating the equations to remove one or more variables.
In our exercise, we focused on eliminating the variable \(x\). By subtracting the first equation \((x + y + z = 4)\) from the second equation \((x + 3y + 3z = 10)\), we successfully removed \(x\), giving us \(y + z = 3\).
Applying this method again between the first and the third equations \((2x + y - z = 3)\), we obtained another equation \(x - 2z = -1\). These resulting equations were simpler and allowed us to find expressions for the remaining variables in terms of one another.

Consistency Check

The consistency check is a crucial step in solving systems of equations. It ensures that a found solution satisfies every original equation. This step is important because it confirms whether the system has any solutions at all and, if it does, whether those solutions are correct.
In our example, after determining the parametric solution \((x, y, z) = (2t - 1, 3 - t, t)\), we substituted these expressions back into the original equations to see if each one held true.

• The first equation: \(x + y + z = 4\)
• The second equation: \(x + 3y + 3z = 10\)
• The third equation: \(2x + y - z = 3\)

By substitution, each equation was satisfied, confirming that the system is consistent and that our parametric form solution is correct.

System of Equations

A system of equations consists of multiple equations that we solve together, considering all the provided conditions. These systems can include any number of equations and variables. In our case, we dealt with three linear equations involving three variables, \(x\), \(y\), and \(z\).
Each equation introduces a new constraint on the possible values of the variables, narrowing down the number of potential solutions. In linear systems such as this one, solutions can be unique, infinite, or nonexistent, depending on the relationships between the equations.
The solution process typically involves using techniques like substitution, elimination, and checking for consistency to find all possible solutions. These methods help in interpreting complex systems and understanding their behavior under different conditions.