Problem 12 - Advanced Engineering Mathematics

Question

In Problems 11 and 12, determine the values of \(x\) and \(y\) for which the matrices are equal. $$ \left(\begin{array}{ll} x^{2} & 1 \\ y & 5 \end{array}\right),\left(\begin{array}{cc} 9 & 1 \\ 4 x & 5 \end{array}\right) $$

Step-by-Step Solution

Answer

The solutions are \((x, y) = (3, 12)\) and \((x, y) = (-3, -12)\).

Step 1: Understand Matrix Equality

For two matrices to be equal, their corresponding elements must be identical. This means that each element in the first matrix must match exactly with the corresponding element in the second matrix.

Step 2: Set Up Equations

Set up equations by equating corresponding elements of the matrices:1. From the first element: \(x^2 = 9\).2. From the second element: The equation \(1 = 1\) is already true.3. From the third element: \(y = 4x\).4. From the fourth element: The equation \(5 = 5\) is already true.

Step 3: Solve Equation 1

Solve the equation \(x^2 = 9\). We find that \(x\) can be either \(3\) or \(-3\) because the square of both values is \(9\).

Step 4: Solve Equation 3 for Each Possible \(x\) Value

Substitute each possible value of \(x\) into the third equation \(y = 4x\):1. If \(x = 3\), then \(y = 4(3) = 12\).2. If \(x = -3\), then \(y = 4(-3) = -12\).

Step 5: Conclusion

The solutions are derived for each possible \(x\) value: - When \(x = 3\), \(y = 12\).- When \(x = -3\), \(y = -12\). These are the pairs that make the matrices equal.

Key Concepts

Solving Equations

When solving equations, the goal is to find the values of unknown variables that make the equation true. In the exercise, we encountered two primary equations derived from the first elements of each matrix. The equation \(x^2 = 9\) required us to find values for \(x\) that satisfy this equality. To solve it:

• We find the square root of both sides, recognizing that \(x\) can be both positive and negative due to squaring.
• Thus, we ultimately find \(x\) can be either \(3\) or \(-3\).

It's important to consider all possible solutions given by the equation, especially when dealing with quadratic expressions like \(x^2 = 9\). Each solution impacts follow-up calculations, like our next steps to solve for \(y\).

Element-wise Comparison

Element-wise comparison is crucial for determining matrix equality. Each corresponding element in one matrix must match the element in the same position in the other matrix. In our exercise, we compared:

• \(x^2\) from the first matrix with \(9\) from the second matrix.
• The constant \(1\) from both matrices observed to be equal.
• \(y\) with \(4x\) derived from corresponding matrix elements.
• The constant \(5\) from both matrices, also equal.

Once each pair of corresponding elements is equated through this method, we can set up and solve equations for unknowns like \(x\) and \(y\). This process ensures thorough checking for equality.

System of Equations

A system of equations refers to a set of equations with multiple variables. Solving them requires finding values for each variable that satisfy all equations simultaneously. In the given exercise:

• We set up two equations: First is \(x^2 = 9\), leading to the solutions \(x = 3\) or \(x = -3\). Second is \(y = 4x\), dependent on the solution from the first equation.
• By substituting back each found \(x\) value into the equation for \(y\), we derive two possible \(y\) values: \(y = 12\) and \(y = -12\).

Each unique \((x, y)\) pair is a solution to this system of equations. Employing these systematic steps ensures that all possible solutions are covered, reflecting the nature of solving simultaneous equations in algebra.