Problem 28 - Precalculus: Mathematics for Calculus

Question

A matrix is given in row-echelon form. (a) Write the system of equations for which the given matrix is the augmented matrix. (b) Use back-substitution to solve the system. $$\left[\begin{array}{rrrrr} 1 & 0 & -2 & 2 & 5 \\ 0 & 1 & 3 & 0 & -1 \\ 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 & -1 \end{array}\right]$$

Step-by-Step Solution

Answer

\(x_1 = 5\), \(x_2 = 2\), \(x_3 = -1\), \(x_4 = -1\).

Step 1: Understand the Augmented Matrix

The given matrix is in row-echelon form and represents an augmented matrix of a system of equations. Each row corresponds to an equation, and the individual elements represent the coefficients and constant term of the equation.

Step 2: Write the System of Equations

Translate each row of the matrix into a corresponding equation. The first row \([1, 0, -2, 2, 5]\) is represented by the equation \(x_1 - 2x_3 + 2x_4 = 5\). The second row \([0, 1, 3, 0, -1]\) translates to \(x_2 + 3x_3 = -1\). The third row \([0, 0, 1, -1, 0]\) becomes \(x_3 - x_4 = 0\). Finally, the fourth row \([0, 0, 0, 1, -1]\) translates to \(x_4 = -1\).

Step 3: Solve for the Fourth Variable

From the fourth equation, \(x_4 = -1\). This value can be substituted back into the other equations to find the remaining variables.

Step 4: Solve for the Third Variable Using Back-Substitution

Substitute \(x_4 = -1\) into the third equation: \(x_3 - (-1) = 0\), which simplifies to \(x_3 = -1\).

Step 5: Solve for the Second Variable Using Back-Substitution

Substitute \(x_3 = -1\) into the second equation: \(x_2 + 3(-1) = -1\), which simplifies to \(x_2 - 3 = -1\) or \(x_2 = 2\).

Step 6: Solve for the First Variable Using Back-Substitution

Substitute \(x_3 = -1\) and \(x_4 = -1\) into the first equation: \(x_1 - 2(-1) + 2(-1) = 5\). Simplifying, we get \(x_1 + 2 - 2 = 5\), which simplifies to \(x_1 = 5\).

Step 7: Present the Solution

The solution to the system of equations is \(x_1 = 5\), \(x_2 = 2\), \(x_3 = -1\), \(x_4 = -1\).

Key Concepts

Augmented Matrix

An augmented matrix is a convenient way to summarize a system of linear equations. It includes all the coefficients of the variables and the constants from the equations. In simple terms, each row in the matrix represents one equation, while the columns represent the coefficients of each variable, plus an additional column for the constants on the right side of the equations.

In the context of matrices, an augmented matrix looks like this:

• The first few columns represent the coefficients of the variables (e.g., coefficients of \(x_1, x_2, x_3\)).
• The final column represents the constants on the other side of the equations.

This structure allows us to visualize and perform operations directly to solve the system of equations. The given problem starts with an augmented matrix that is already in row-echelon form and helps us convert it back into its corresponding system of equations by interpreting each row as an equation.

Row-Echelon Form

The row-echelon form is a certain kind of matrix shape that is particularly useful in solving systems of linear equations. This form makes the process of solving the equations much more straightforward by setting the matrix up like a staircase:

• Each leading entry (the first non-zero number from the left) in a row is to the right of the leading entry in the row above.
• All rows with all zero entries, if any, are at the bottom of the matrix.

For example, in the provided matrix, each leading coefficient steps over to the right as you move down the rows. This ensures that no two leading coefficients are aligned vertically, providing a clear path to decipher the values of the variables step-by-step using back-substitution. This form is crucial because it prepares us for the next phase: simplifying equations to find precise solutions.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. Solving the system means finding values for the unknowns that satisfy all equations simultaneously. The process generally involves:

• Writing down the system exactly as represented by the augmented matrix.
• Interpreting each row of the matrix as an individual equation based on the coefficients and constants.

In our exercise, each of the four rows in the matrix is translated into a linear equation that involves variables \(x_1\) through \(x_4\). With row-echelon form and the values identified through back-substitution, solving these equations becomes much simpler. By working from the bottom row to the top, we substitute known values back into the previous equations to solve for unknowns step by step. This method ensures that each variable is accurately determined, leading to a clear and correct solution for the entire system.