Problem 34 - College Algebra

Question

write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{rr} 3 & 0 \\ -3 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{r} 6 \\ -7 \end{array}\right] $$

Step-by-Step Solution

Answer

The system of linear equations for the given matrix equation is x = 2 and y = -1.

Step 1: Identification of Matrices

First, let's identify our matrices. We have three matrices here. Let's call the first matrix with elements, [3 0; -3 1] as Matrix A, [x; y] as Matrix X and [6; -7] as Matrix B. So our equation becomes AX=B.

Step 2: Matrix Multiplication

Now we perform the multiplication of matrix A by matrix X, that is [3 0; -3 1] * [x; y], to get a new matrix. The multiplication results in a new matrix, let's call it Matrix C: [3x+0y; -3x+1y] = [3x; -3x+y].

Step 3: Writing linear equations

The equation AX=B now becomes [3x; -3x+y] = [6; -7], this implies two system of linear equations. First row equal to first row, 3x = 6, which can be further simplified to x = 2. Second row equal to second row, -3x+y = -7. Substituting x = 2 into the second equation we get -3(2) + y = -7, which can be simplified to y = -1.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves calculating the product of two matrices, where the number of columns in the first matrix must be equal to the number of rows in the second matrix.

• In our example, Matrix A is a 2x2 matrix with elements \([3, 0; -3, 1]\) and Matrix X is a 2x1 matrix \([x; y]\).
• This means we can multiply Matrix A by Matrix X, resulting in a new matrix with elements \([3x+0y; -3x+1y]\), which simplifies to \([3x; -3x+y]\).

Matrix multiplication is not as straightforward as multiplying numbers; it requires taking the dot product of the rows of the first matrix with the columns of the second matrix.
This highlights the importance of following the order of operations correctly to ensure the correct result.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces.
It provides powerful tools to solve systems of linear equations efficiently. Matrix notation is particularly useful because it allows for more compact representation and computation.

• For instance, instead of writing multiple individual linear equations, you can write a single matrix equation \(AX = B\).
• This not only simplifies the way we handle equations but also enables numerous calculations, like finding determinants, inverses, and computing eigenvalues or eigenvectors.

Linear algebra is widely applicable in various fields, including computer science, engineering, physics, and economics, making it an essential topic to master.

Solving Linear Systems

Solving a system of linear equations is one of the primary applications of linear algebra. It involves finding the values of variables that satisfy all equations simultaneously.
To solve our example system, we can write the matrix equation \(AX = B\) as two separate linear equations:

• The first equation from \([3x; -3x+y] = [6; -7]\) is \(3x = 6\). This can be quickly solved to get \(x = 2\).
• The second equation is \(-3x + y = -7\). By substituting \(x = 2\), we find \(-3(2) + y = -7\), simplifying to \(y = -1\).

This means the solution to the linear system is \(x = 2\) and \(y = -1\).
Understanding how to manipulate and solve these equations is crucial, especially when dealing with real-world scenarios that can be modeled using linear relationships.