Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system of linear equations and check any solution algebraically. $$\left\\{\begin{array}{rr} -x+3 y+z= & 4 \\ 4 x-2 y-5 z= & -7 \\ 2 x+4 y-3 z= & 12 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 0\), \(y = 1\), and \(z = 1\).

Step 1: Raw Operation

Multiply the first equation by \(4\) and the second equation by \(1\), and then subtract the first equation from the second to derive a new system of equations: \[ \begin{cases} -4x+12y+4z = 16 \ 4x-2y-5z= -7\ 2x+4y-3z =12 \end{cases} \] Adding equation \(1\) and equation \(2\), we get \(10y - z = 9\). Now we have a new system of equations: \[ \begin{cases} -x+3y+z = 4\ 10y - z = 9\ 2x+4y-3z =12 \end{cases} \]

Step 2: Solve for \(z\)

From equation \(2\), it can be derived that \(z = 10y - 9\). Substitute \(z\) into equation \(1\) and we get \(-x + 3y + (10y - 9) = 4\), which simplifies to \(x = 13y - 13\). Now, we have the following system of equations: \[ \begin{cases} z = 10y - 9\ x = 13y - 13 \end{cases} \]

Step 3: Substitute \(x\) and \(z\) into equation \(3\)

Substitute \(x = 13y - 13\) and \(z = 10y - 9\) into equation \(3\), we get \(2(13y - 13) + 4y - 3(10y - 9) = 12\), simplifying this equation leads to \(6y = 6\), thus \(y = 1\).

Step 4: Solve for \(x\) and \(z\)

Substitute \(y = 1\) into \(x = 13y - 13\) and \(z = 10y - 9\), we get \(x = 0\) and \(z = 1\).

Step 5: Verify the Solution

Plug \(x = 0\), \(y = 1\), and \(z = 1\) back into the original equations to verify that they satisfy all of them. The system of equations is true, hence the solution is correct.

Key Concepts

Gaussian Elimination

Gaussian Elimination is a method used to solve systems of linear equations. It involves a series of operations to transform a matrix into its row-echelon form, where you can easily solve for the unknowns. The first step is to align the equations and gradually eliminate coefficients below the pivot positions using row operations like swapping, multiplying, and adding equations together. In the given exercise, the equations of the system are:

• \(-x + 3y + z = 4\)
• \(4x - 2y - 5z = -7\)
• \(2x + 4y - 3z = 12\)

To eliminate variables and simplify the system, the first equation is multiplied by 4 and subtracted from the second equation, leading to the simplification of terms and forming a new system of equations. Through these systematic row operations, Gaussian Elimination aims to simplify the system until solutions become straightforward.

Substitution Method

The Substitution Method is a straightforward technique in solving systems of equations. Once a variable is solved in terms of the others, substitute it back into the remaining equations. In our problem, once the transformed set of equations is established through Gaussian elimination, we solve for one variable using another. Specifically, from \(10y - z = 9\), derive \(z = 10y - 9\) and replace \(z\) in the other equations. Also, from \(-x + 3y + (10y - 9) = 4\), solve for \(x\) as \(x = 13y - 13\). These expressions allow substitution into the remaining equations. This breakdown simplifies the system, step by step, leading to finding the values of all variables.

Algebraic Verification

Once the solution is found, it is crucial to verify it algebraically. This is to ensure that the values obtained satisfy all original equations. In our example, the solutions obtained are \(x = 0\), \(y = 1\), and \(z = 1\). By substituting these values back into the original equations:

• First Equation: \(-0 + 3(1) + 1 = 4\)
• Second Equation: \(4(0) - 2(1) - 5(1) = -7\)
• Third Equation: \(2(0) + 4(1) - 3(1) = 12\)

Each one holds true, indicating that the process and the derived solutions are accurate. Algebraic verification is a critical step in solving linear equations as it confirms the correctness of the obtained solution.

Solution Checking

After finding a solution to a system of equations, it's not only about having the answer but ensuring its correctness. Solution checking involves substituting the obtained values back into the original system of equations to confirm they are indeed correct solutions. Let's do a solution check for \(x = 0\), \(y = 1\), \(z = 1\): substitute these into all equations of the system:

• Equation 1: Check if \(-0 + 3(1) + 1 = 4\)
• Equation 2: Check if \(4(0) - 2(1) - 5(1) = -7\)
• Equation 3: Check if \(2(0) + 4(1) - 3(1) = 1\)

All checks confirm the left and right sides of each equation are equal, verifying that the solution is consistent and complete. This reaffirms trust in the entire problem-solving method employed.