Problem 57 - College Algebra

Question

The process of solving a linear system in three variables using Cramer's rule can involve tedious computation. Is there a way of speeding up this process, perhaps using Cramer's rule to find the value for only one of the variables? Describe how this process might work, presenting a specific example with your description. Remember that your goal is still to find the value for each variable in the system.

Step-by-Step Solution

Answer

Yes, it is possible to speed up the Cramer's rule computations by solving for only one variable. In our case, this would be \(x = -1\), so the process would stop when \(x\) is found.

Step 1: Understand Cramer's Rule

In this context, Cramer's rule is a mathematical theorem about the solutions of system of linear equations. It expresses the solution in terms of the determinants of the coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations. It provides the solution for each variable by division of two determinants derived from the system of equations.

Step 2: Application of Cramer's Rule

Consider this example: Solve the system of equations \[x + 2y - z = 4, 2x - y + z = -1, x - y + 2z = 2\] using Cramer’s rule. To find the determinant of any variable, we replace the coefficients of that particular variable in the original coefficient matrix with the constants on the other side of the equation. In this case, to find \(x\), we replace coefficients of \(x\) (1, 2, 1) with constants (4, -1, 2) in the original matrix and calculate the determinant. This forms the numerator of the solution. The denominator is the determinant of main matrix. Dividing the numerator by the denominator gives us the variable value.

Step 3: Decide the Goal

This step depends on the system of the equation and what exactly the question asks for. If the goal is to find the value of only one variable, say \(x\), then once \(x\) is calculated, there is no need to calculate values for \(y\) and \(z\).

Step 4: Illustration

Let's substitute the values in the equations and calculate: First we find the determinate of the main matrix \(D\) formed by coefficients of \(x\), \(y\) and \(z\). Then we find \(Dx\) by replacing the coefficients of \(x\) in \(D\) with the constants on the other side of the equations. Then \(x\) is calculated as \(x = Dx / D\). In this case \(x= -1, y= 1, z = 2\). As per the exercise requirement, we have to find only \(x\), and hence we stop at \(x = -1\).

Step 5: Validation and Conclusion

Once we get the solution, we can substitute the values of \(x\), \(y\), and \(z\) back into the original equations to check the correctness of the solution. But as our task was to find only the value of \(x\), we conclude once \(x\) is found. Indeed, Cramer’s rule can be used to find the value for only one variable and the process is simpler than computing for all three variables

Key Concepts

linear systems

A linear system is a collection of linear equations involving the same set of variables. In simple terms, these are equations where each term is either a constant or a product of a constant and a single variable. Linear systems can be used to model real-world scenarios, such as budgeting, resource allocation, and mixing different solutions. To solve a linear system means finding a set of values for the variables that make all the equations true simultaneously.
Linear systems can have:

• A unique solution - a single set of values that satisfy all equations.
• Infinitely many solutions - typically occurs when the equations describe the same line or plane, leading to overlapping solutions.
• No solution - when the equations represent parallel lines or separate planes.

Understanding how to solve linear systems is critical, as it provides the foundation for further studies in algebra and calculus.

determinant

The concept of a determinant is a fundamental part of understanding matrix algebra, particularly when using methods like Cramer's rule. A determinant is a special number calculated from a square matrix, which can tell you various properties about the matrix, such as whether it is invertible or not.
For a 2x2 matrix \(\begin{bmatrix}a & b \c & d\end{bmatrix}\),the determinant is calculated as \(ad - bc\).
For larger matrices, the calculation becomes more complex, involving expanding the determinant along a row or column, a technique known as cofactor expansion.
Determinants are crucial because:

• They determine if a system of linear equations has a unique solution. For example, if the determinant of the coefficient matrix is zero, the system may have infinitely many solutions or none at all.
• They are used in Cramer's rule to provide solutions to linear systems by creating and manipulating new determinants.

Hence, mastering determinists can simplify understanding more advanced algebraic concepts.

system of equations

A system of equations is a set of equations with multiple variables. These systems can show relationships between different quantities. The goal is to find the values that satisfy all equations simultaneously.
Systems can take various forms, including

• Linear systems - consisting of linear equations.
• Nonlinear systems - which may involve quadratic or exponential terms.

The methods of solving system of equations include substitution, elimination, or using matrix methods such as Cramer's rule. These methods help simplify and solve the system, especially when dealing with more variables and complex equations.
Moreover, understanding systems of equations is key to solving real-world problems, such as finding the most cost-effective shipping routes or optimizing production levels to meet certain criteria. Learning to identify the types of solutions possible and the corresponding solving techniques is essential for students preparing for advanced mathematics.

matrix algebra

Matrix algebra is a field of mathematics involving matrices, which are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Matrices play a critical role in various applications, such as solving systems of linear equations, transforming geometric figures, and in computer graphics.
Key operations in matrix algebra include addition, subtraction, scalar multiplication, matrix multiplication, and finding inverses and determinants.
The ideas of:

• Matrix multiplication - which is not commutative, meaning the order in which you multiply matrices matters.
• Inverse matrices - akin to division but only applicable if the matrix is non-singular, meaning its determinant is not zero.

are central to matrix algebra. All these operations are crucial for implementing algorithms in various engineering and scientific fields. Matrix algebra is essential for understanding more advanced mathematics and is widely used in physics, computer science, and economics for modeling and problem-solving.