Problem 37 - Algebra 2

Question

Find the value of each variable. $$ \left[\begin{array}{cc}{x} & {y-2} \\ {z} & {w+4}\end{array}\right]+\left[\begin{array}{rr}{2} & {5} \\ {-2} & {4}\end{array}\right]=\left[\begin{array}{ll}{6} & {1} \\ {4} & {8}\end{array}\right] $$

Step-by-Step Solution

Answer

x = 4, y = -4, z = 6, w = 0

Step 1: Add corresponding entries

To find the values of the variables, first add the corresponding entries of the two matrices.

Step 2: Set up the equations

From the addition in step 1, create an equation for each position in the matrices that includes a variable.

Step 3: Solve for x

From the first equation, solve for the variable x.

Step 4: Solve for y

From the second equation, solve for the variable y.

Step 5: Solve for z

From the third equation, solve for the variable z.

Step 6: Solve for w

From the fourth equation, solve for the variable w.

Key Concepts

Solving Systems of Equations

Understanding how to solve systems of equations is critical in algebra and applies to various disciplines. It's essentially about finding the values of unknowns that satisfy multiple equations simultaneously.

When it comes to matrix addition, solving for variables within a system becomes a matter of aligning equations derived from the sum of corresponding matrix elements. As in the exercise above, after adding corresponding entries of two matrices, we obtain several algebraic equations that represent a system.

Each variable within the resulting matrix is isolated in the following steps to find the unique values that satisfy the original matrix equation. This step-by-step process mirrors traditional methods for solving systems such as substitution or elimination. The key is to carefully form equations based on the sum of matrix elements and then solve for the variables like you would solve any other linear system.

Algebraic Expressions

Algebraic expressions form the foundation of most algebraic problems. They consist of variables, constants, and operating symbols that express operations to be performed. As seen in matrix addition, variables such as 'x', 'y', 'z', and 'w' are involved, combined with numerical constants.

It's imperative to understand how to manipulate these expressions to isolate and solve for variables, which is the heart of algebra. In the matrix addition exercise, after equating the sum of matrices with the resultant matrix, you'll get a set of algebraic expressions. Solving them requires basic algebraic skills—adding or subtracting constants, and isolating variables.

Ensuring the correct application of algebraic operations allows for the seamless transition from a complex looking matrix equation to a straightforward system of equations that can be solved with familiar algebraic techniques.

Matrix Operations

Matrix operations, including addition, are essential in solving multidimensional problems in linear algebra. Matrices are grids of numbers or expressions that follow specific rules for addition, subtraction, multiplication, and division (inverse multiplication).

For matrix addition, like in our exercise, the operation is quite straightforward. You simply add corresponding elements from each matrix to form a new matrix. However, it's important to note that matrix operations only make sense when the dimensions of the matrices are the same, or satisfy certain criteria for multiplication.

In practice, matrix operations can represent and solve complex systems of equations, transform geometric figures, handle data in computer graphics, and more. The beauty of matrices is their ability to compactly represent numerous equations or data points, making it possible to efficiently work with large systems.