Problem 6 - Algebra and Trigonometry

Question

Find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{l} {x} \\ {7} \end{array}\right]=\left[\begin{array}{l} {11} \\ {y} \end{array}\right] $$

Step-by-Step Solution

Answer

The solution is \(x = 11\) and \(y = 7\).

Step 1: Equalize first elements of the matrices

Equalize the first elements, we then have \(x = 11\)

Step 2: Equalize second elements of the matrices

Equalize the second elements, \(7 = y\) or \(y = 7\)

Key Concepts

Variable Solving in Matrices

To find the values of variables that make matrices equal, we focus on solving for each variable separately. This comes from the idea that two matrices are equal if and only if their corresponding elements are equal. This means that we treat each position like an independent equation.

• First, identify each variable in the matrices. For a simple matrix like the one given, you only have a couple variables like \(x\) and \(y\).
• Next, write an equation for each corresponding pair of elements: for example, \(x = 11\) from the first element of both matrices.
• Finally, solve these equations as you would any regular algebraic equation. This is straightforward when the problem is simple, like in step 1, where \(x\) is exactly equal to 11, and in step 2 for \(y = 7\).

It's crucial to ensure all elements match to confirm the matrices are truly equal.

Understanding Matrices

Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. They appear in all sorts of mathematical problems, especially in linear algebra. A matrix can represent a variety of information or even systems of equations.

• Matrices are notated by square brackets with elements placed according to their row number and column number. For example, a matrix with 2 rows and 2 columns is a 2x2 matrix.
• The position of an element in a matrix is crucial. Two matrices can only be equal if they have the same dimensions and their corresponding elements are equal.
• Matrices are widely used in computer graphics, statistics, and engineering to model complex interactions.

A simple exercise like taking equal matrices allows for practicing how to navigate through these elements.

Linear Equations from Matrices

Linear equations are equations that make a straight line when graphed. They have many forms, including being part of matrices, especially when solving systems of equations. From matrices, linear equations can be easily extracted:

• Each element in a row of a matrix can represent a part of a linear equation. When two matrices are set to be equal, each corresponding row and column provide a linear equation.
• For example, from the solution \(x = 11\), this can be viewed as a linear equation where there's just a single variable, making it a straight line at \(x = 11\).
• Matrix equality exercises train your mind to convert these into manageable algebraic problems, essentially creating a series of smaller, easily solvable linear equations.

Converting matrix problems into linear equations is a powerful method for organizing and solving mathematical problems.