Solving Matrix Equations: Finding X And Y

Hey guys! Let's dive into a cool math problem where we'll figure out the values of x and y that make a matrix equation true. This is a great exercise in linear algebra, and it's super important for all sorts of applications, from computer graphics to engineering. Ready to get started?

Understanding the Problem: Matrix Addition and Scalar Multiplication

Alright, so the core of our problem is a matrix equation. We've got two matrices being added together, and that sum is equal to a scalar (a single number) multiplied by another matrix. Let's break down each part to make sure we're all on the same page. Remember, understanding the basics is key to solving more complex problems!

Firstly, we have matrix addition. When you add matrices, you simply add the corresponding elements. For example, if you have two 2x2 matrices, you add the top-left element of the first matrix to the top-left element of the second matrix, and so on. The matrices must have the same dimensions for addition to be possible. Secondly, we have scalar multiplication. This is when you multiply a matrix by a single number (the scalar). You multiply each element of the matrix by that scalar. For example, if you multiply a 2x2 matrix by the scalar 3, you multiply each of the four elements by 3. Also remember that in the matrix world, the order of operations matters. Just like in regular arithmetic, we perform multiplication before addition (unless parentheses tell us otherwise).

Let's refresh our knowledge of the matrices. A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In our case, we're dealing with 2x2 matrices, meaning they have two rows and two columns. Each element in the matrix has a specific position, identified by its row and column number. This structured organization is what allows us to perform operations like addition and multiplication.

The Equation

Here's the equation we're working with:

$egin{pmatrix} 5 & 15x 1 & x

egin{pmatrix} 1 & y+3 2 & 1

= 3 egin{pmatrix} 1 & 2 -1 & 1

$egin{pmatrix} 0 & 2 1 & 3

As you can see, we have matrices with variables in them (x and y). Our goal is to manipulate this equation using the rules of matrix algebra until we can isolate x and y and find their values. It's like a puzzle, and we're the detectives! We are going to use some important concepts that will helps us solve this problem: matrix addition and scalar multiplication.

Step-by-Step Solution: Unraveling the Matrix Equation

Okay, let's roll up our sleeves and solve this equation step by step. We'll start with the left side of the equation and then tackle the right side. Our strategy is to simplify each side as much as possible, which will allow us to create a new, simpler equation. We'll then extract the necessary information from this simpler equation to find the values of x and y. Remember to be patient and careful with each step. Mistakes are easy to make, but they are also easy to fix if you're methodical.

Step 1: Matrix Addition on the Left Side

First, we'll perform the addition of the two matrices on the left side of the equation. Remember, to add matrices, you add the corresponding elements. So:

$egin{pmatrix} 5 & 15x 1 & x

egin{pmatrix} 1 & y+3 2 & 1

= egin{pmatrix} 5+1 & 15x+(y+3) 1+2 & x+1

= egin{pmatrix} 6 & 15x+y+3 3 & x+1

See? It's pretty straightforward. We've simplified the left side of the equation into a single matrix. Now, let's move on to the right side.

Step 2: Matrix Multiplication and Scalar Multiplication on the Right Side

Now we'll deal with the right side of the equation. We have a scalar multiplication (the '3') and a matrix multiplication. Remember to do the matrix multiplication first. Here’s how it works:

$egin{pmatrix} 1 & 2 -1 & 1

$egin{pmatrix} 0 & 2 1 & 3

= egin{pmatrix} (10)+(21) & (12)+(23) (-10)+(11) & (-12)+(13)

= egin{pmatrix} 0+2 & 2+6 0+1 & -2+3

= egin{pmatrix} 2 & 8 1 & 1

Now, we multiply the resulting matrix by the scalar 3:

$3 * egin{pmatrix} 2 & 8 1 & 1

= egin{pmatrix} 32 & 38 31 & 31

= egin{pmatrix} 6 & 24 3 & 3

Great! We've simplified the right side to a single matrix as well.

Step 3: Equating the Matrices

Now, our equation looks like this:

$egin{pmatrix} 6 & 15x+y+3 3 & x+1

= egin{pmatrix} 6 & 24 3 & 3

For two matrices to be equal, their corresponding elements must be equal. This gives us a system of equations. Let's create these equations by comparing the elements:

• Element (1,1): 6 = 6 (This is already true, so it doesn't help us find x or y)
• Element (1,2): 15x + y + 3 = 24
• Element (2,1): 3 = 3 (Again, this is true, but not helpful)
• Element (2,2): x + 1 = 3

Step 4: Solving for x and y

From the equation x + 1 = 3, we can easily solve for x: x = 3 - 1 x = 2

Now that we know x = 2, we can substitute it into the other equation, 15x + y + 3 = 24, to solve for y: 15(2) + y + 3 = 24 30 + y + 3 = 24 y + 33 = 24 y = 24 - 33 y = -9

Step 5: The Final Answer

Therefore, the values that satisfy the original equation are x = 2 and y = -9.

Conclusion: Mastering Matrix Equations

And there you have it, guys! We've successfully solved the matrix equation and found the values of x and y. This is a fundamental concept in linear algebra, and hopefully, you found this explanation helpful. Keep practicing, and you'll become a matrix master in no time! Remember, the key is to break down the problem into smaller, manageable steps. Also, be careful with your arithmetic, and always double-check your work. You've got this!

If you have any questions or want to try another matrix problem, just let me know. Happy calculating!