Problem 16 - College Algebra with Modeling and Visualization

Question

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rr|r} 1 & 4 & -2 \\ 0 & 1 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

(-14, 3)

Step 1: Identify the System of Equations

The augmented matrix represents the following system of equations: 1. \( x + 4y = -2 \) 2. \( y = 3 \)

Step 2: Solve the Second Equation

Since the second equation is already solved for \( y \), we can substitute \( y = 3 \) directly into the equation. This gives us the value of \( y \) needed for substitution into the first equation.

Step 3: Substitute to Solve for x

Using the value of \( y = 3 \) from the second equation, substitute into the first equation: \( x + 4(3) = -2 \)Simplify and solve for \( x \):\( x + 12 = -2 \)Subtract 12 from both sides: \( x = -2 - 12 \)\( x = -14 \)

Step 4: Write the Solution as an Ordered Pair

The solution to the system is the ordered pair \((x, y) = (-14, 3)\). Since this is a system with two variables, our final answer should be in the form of an ordered pair.

Key Concepts

Augmented Matrix

An augmented matrix is a crucial concept in linear algebra, used to represent a system of linear equations in a simplified way. It combines the coefficients of the variables and the constants directly from the equations into a single matrix setup. This form makes it easier to perform operations and visualize the steps needed to solve the equations.

For example, consider the system of equations derived from the given augmented matrix:

• Equation 1: \( x + 4y = -2 \)
• Equation 2: \( y = 3 \)

In augmented matrix form, these are compacted into:\[\left[\begin{array}{rr|r}1 & 4 & -2 \0 & 1 & 3 \\end{array}\right]\]The vertical line separates the coefficients of the variables \(x\) and \(y\), from their corresponding constants. This format is suitable for applying methods like Gaussian elimination to solve the system.

Row-Echelon Form

Row-echelon form is a type of matrix representation where each leading entry (a non-zero number at the beginning of a row) is to the right of the leading entry in the previous row. This helps to systematically solve linear systems of equations by simplifying the matrix using operations.

• The leading entry in each row must be 1, if possible (though this is more accurately the case in a reduced row-echelon form).
• All entries below the leading entry must be zero.
• If a row contains only zeros, it is positioned at the bottom of the matrix.

Our matrix is already in a basic row-echelon form:\[\left[\begin{array}{rr|r}1 & 4 & -2 \0 & 1 & 3 \\end{array}\right]\]The system is simplified so that we can easily use back substitution to find the solutions for \(x\) and \(y\). It sets us up perfectly for the next step in solving the linear equations.

Back Substitution

Back substitution is the method used to solve equations once a system has been transformed to row-echelon form. With the second equation already solved for \(y\), we can find \(x\) by substituting back into the first equation.

• Start with the last equation in the system, which has been reduced to a basic form.
• Use the already known values to solve for the remaining variables.

In our example:- Equation 2: \( y = 3 \) gives us the value of \( y \).- Substitute \( y = 3 \) into Equation 1: \( x + 4(3) = -2 \).- This results in \( x + 12 = -2 \).- Solve for \( x \): \( x = -14 \).Thus, the final solution of the system is an ordered pair \((-14, 3)\). This backward approach ensures that values unfold logically, leading to an accurate set of solutions.