Problem 11 - Advanced Engineering Mathematics

Question

In Problems 11 and 12, determine the values of \(x\) and \(y\) for which the matrices are equal. $$ \left(\begin{array}{rr} 1 & x \\ y & -3 \end{array}\right),\left(\begin{array}{cc} 1 & y-2 \\ 3 x-2 & -3 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \(x = 2\) and \(y = 4\).

Step 1: Understand Matrix Equality

Two matrices are equal if and only if all their corresponding elements are equal. We will use this concept to find the values of \(x\) and \(y\) for which the given matrices are equal.

Step 2: Set Up Equations from Matrix Elements

Compare corresponding elements of the matrices to form equations: \(1 = 1\), \(x = y - 2\), \(y = 3x - 2\), and \(-3 = -3\). Simplifying, we focus on valid equations: \(x = y - 2\) and \(y = 3x - 2\).

Step 3: Solve the Equations Simultaneously

Substitute \(y = x + 2\) from \(x = y - 2\) into \(y = 3x - 2\). This gives \(x + 2 = 3x - 2\). Simplify to find \(4 = 2x\), therefore, \(x = 2\).

Step 4: Find the Value of y Using x = 2

Substitute \(x = 2\) back into \(y = x + 2\) to find \(y = 2 + 2 = 4\).

Key Concepts

Simultaneous Equations

Simultaneous equations involve finding values for unknown variables that satisfy multiple equations at the same time. In this exercise, the goal was to identify values of \(x\) and \(y\) such that both equations derived from the matrix elements are true.
We started with:\[x = y - 2\] and \[y = 3x - 2\].
Solving them simultaneously helps to find a common solution for both. To begin, you substitute one equation into the other, which is a logical way to fuse both conditions and find values that satisfy the whole system.

When approached step-by-step:

• Choose one equation to express one variable in terms of the other.
• Substitute back into the second equation to find a specific numeric value for the variables.

In practice, solving simultaneous equations requires meticulous substitution and simplification steps to ensure all possible solutions are examined.

Matrix Elements

Matrix elements are the individual items or numbers found within a matrix, represented as a grid. By definition, matrices are equal when each corresponding element matches exactly. In this case, the matrices are:
\[\begin{pmatrix} 1 & x \ y & -3 \end{pmatrix}, \begin{pmatrix} 1 & y-2 \ 3x-2 & -3 \end{pmatrix}\]

Key points to consider are:

• The element in the top left (1) corresponds to the same position in both matrices, and as seen, they match perfectly.
• Differences or unknowns in the other areas provide the necessary clues and equations to solve. For instance, \(x\) corresponds to \(y-2\).
• Examining each element positions guide us to what we need to compare and ultimately prove equality.

By analyzing these elements, one can effectively construct equations to determine variables.

Substitution Method

The substitution method is a way to simplify and solve simultaneous equations. By isolating one variable in terms of the other from one equation, you can replace this expression wherever that variable appears in the other equation. Begin by rearranging an equation to find an expression for one of the variables. For instance:
From \(x = y - 2\), you can express \(y\) in terms of \(x\) as \(y = x + 2\).

Here's how you proceed:

• Select an equation that is easy to isolate a variable.
• Substitute this expression into the other equation to obtain a single-variable equation.
• Solve this new equation for its variable.

In this case, by substituting into \(y = 3x - 2\), you simplify the process to only deal with one unknown at a time.
This method is effective because it reduces complex systems into more manageable, linear steps.