Problem 35 - Algebra and Trigonometry

Question

Solve for \(x\) and \(y\) $$ \left[\begin{array}{cc}{x} & {2 y} \\ {4} & {6}\end{array}\right]=\left[\begin{array}{cc}{2} & {-2} \\ {2 x} & {-6 y}\end{array}\right] $$

Step-by-Step Solution

Answer

\( x = 2, y = -1 \)

Step 1: Equate corresponding elements

Start by equating the corresponding elements of the two matrices, since for matrices to be equal, all their corresponding elements must be equal. This gives us two equations: 1.1: \( x = 2 \) 1.2: \( 2y = -2 \) 1.3: \( 4 = 2x \) 1.4: \( 6 = -6y \)

Step 2: Solve for x from the first equation

From equation 1.1 from Step 1, solve for \( x \), which is already simplified: \[ x = 2 \]

Step 3: Verify x with third equation

Using the third equation (1.3) from Step 1, verify the solution for \( x \) to ensure consistency: Substitute \( x = 2 \) into the equation:\[ 4 = 2x \rightarrow 4 = 2(2) \rightarrow 4 = 4 \]This confirms that \( x = 2 \) is correct.

Step 4: Solve for y from the second equation

Using equation 1.2 from Step 1, solve for \( y \):\[ 2y = -2 \rightarrow y = \frac{-2}{2} \rightarrow y = -1 \]

Step 5: Verify y with fourth equation

Using equation 1.4 from Step 1, verify the solution for \( y \):Substitute \( y = -1 \) into the equation:\[ 6 = -6y \rightarrow 6 = -6(-1) \rightarrow 6 = 6 \]This confirms that \( y = -1 \) is correct.

Key Concepts

Equivalent Matrices

Matrix equations involve working with two matrices that are set to be equal to each other. This concept revolves around "equivalent matrices." When two matrices are equivalent, all their corresponding elements must match. This means each element in one matrix has a direct Counterpart in the other matrix with the exact same value.
To understand this better, think of two matrices like identical twin siblings. They must be alike in every way - from hair down to the toes. Just like that, for two matrices to be equivalent, we have to ensure:

• Every element from the first row and first column of the first matrix equals the element from the first row and first column of the second matrix.
• Every element from the second row and second column of the first matrix equals the element from the second row and second column of the second matrix.

Once you have this matching, you can then establish equations for each pair of corresponding elements which will lead you to solve for unknowns.

Solving Systems of Equations

When you have a matrix equation like this one, solving it is much like solving a system of equations.
Each pair of matching elements represents an equation. So if you had a matrix of size 2x2, you would actually have four equations to work with from aligning the individual elements.
These systems of equations can sometimes look daunting, but they allow us to uncover values for unknowns like "x" and "y" in a step-by-step manner. Here’s how you approach it:

• Create equations from the elements corresponding to the two matrices: start from the top-left and move to the bottom-right.
• For each matrix element, ensure you have an equivalent equation to set up and work with.
• Combine these equations to isolate and solve for one unknown.

It’s like detective work: you look at the different clues (or equations) and use them to piece together the answers.

Matrix Equality

Matrix equality is a foundational principle when dealing with matrices. Two matrices are said to be equal when their respective elements in each position are identical.
Imagine two boxes filled with toys. Each box must have the same toy at the same position. If both have a teddy bear in the first position, they are equal; else, they aren't.
When given a matrix equality problem:

• Ensure the matrices are the same size. You can only compare matrices element-wise if they’re the same dimension.
• Double-check each corresponding element since one mismatch means the matrices aren’t equal.

For equations like in our example, understanding matrix equality simplifies the process as it guides us in forming reliable equations to define our unknowns.

Steps in Solving Matrix Equations

Solving matrix equations might initially seem baffling, but when broken down, it’s a sequence of logical, manageable steps. Here is an easy-to-follow process:

• Equate Elements: Begin by comparing each corresponding element from the given matrices. This will give you a set of equations to work with.
• Isolate Variables: For each equation, solve for the unknown variables. Typically, start with the simplest equation first.
• Verify Solutions: Check the solution back in their respective positions to ensure they satisfy each matrix equation. Verifying helps catch any errors before accepting the solutions.

Breaking down the problem this way, you'll find that solving for unknowns can be satisfying. It's like finishing a puzzle piece by piece, and soon enough, the complete picture becomes clear.