Problem 33 - College Algebra

Question

write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{cc} 4 & -7 \\ 2 & -3 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{r} -3 \\ 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The system of linear equations is: 4x - 7y = -3 and 2x - 3y = 1

Step 1: Understand Matrix Multiplication

In our case, matrix A is multiplied by matrix X. Matrix multiplication follows a specific rule. If A= [a ,b] and X= [x,y], then the product AX would be [ax + by]. This rule should be applied separately for each row of matrix A.

Step 2: Apply Matrix Multiplication

Let's apply the matrix multiplication for each row of matrix A. The equations become: 4x - 7y = -3 and 2x - 3y = 1. These two equations form the system of linear equations which is the answer.

Key Concepts

Matrix Multiplication

Matrix multiplication is a process of combining rows and columns of two matrices to produce a new matrix. It's fundamentally different from regular arithmetic multiplication. In this process, we take each row from the first matrix and each column from the second matrix and compute the sum of the products of their corresponding elements. This general rule must be applied consistently:

• Take the elements of the row from the first matrix.
• Pair them with elements of the column from the second matrix, in order.
• Multiply each pair and add the results together.

When you do this for a matrix equation like the one given, it means computing each element in the resulting matrix by applying these rules to every combination of rows and columns. This is what leads to our system of equations. This method allows us to simplify or solve equations in linear algebra.

Systems of Equations

A system of equations is a set of equations with multiple variables. Each equation in the system shares these variables, and the goal is to find values for each variable that satisfy all equations simultaneously. These systems can often be solved using various methods such as substitution, elimination, or matrix operations such as the one shown in the example.

For instance, in the context of matrix equations, converting them into system of equations can make them easier to solve by traditional algebraic means. By breaking one matrix equation into two simple linear equations, you turn a complex problem into a manageable one. In the example provided:

• You end up with the two equations: \(4x - 7y = -3\) and \(2x - 3y = 1\).
• This forms a system where each equation must be satisfied simultaneously.

With each equation affecting the same variables, checking their intersections often reveals the solution to the system.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operation symbols such as addition, subtraction, multiplication, and division. These expressions can represent problems and solutions by showing relationships between different quantities. In the context of linear equations, algebraic expressions form the basis of equations and systems, as they define the constraints and solutions.

In our example, the transformation from a matrix to a system leads to two expressions:

• The expression \(4x - 7y = -3\)
• The expression \(2x - 3y = 1\)

Both are algebraic expressions that involve variables \(x\) and \(y\) and constants. Algebraic expressions are crucial because they allow us to communicate complex mathematical ideas in a structured, solvable form. By solving these expressions, we find the values of \(x\) and \(y\), which satisfy both equations, thus solving the original equation in a simpler way.