Matrix Equation: Find The Value Of 2x - Y

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Hey guys! Today, we're diving into a fun little matrix equation problem. We're given an equation involving matrices, and our goal is to figure out the value of 2x - y. Sounds like a plan? Let's jump right in!

Understanding the Problem

Before we start crunching numbers, let's make sure we understand the problem inside and out. We have a matrix equation that looks like this:

3(5xy4)+(−6−43−y−7)=(98135)3\begin{pmatrix} 5 & x \\ y & 4 \end{pmatrix} + \begin{pmatrix} -6 & -4 \\ 3-y & -7 \end{pmatrix} = \begin{pmatrix} 9 & 8 \\ 13 & 5 \end{pmatrix}

Our mission, should we choose to accept it (and we do!), is to find the values of x and y that make this equation true. Once we have those values, we can easily calculate 2x - y. This involves the concepts of matrix scalar multiplication and matrix addition, which are fundamental in linear algebra. Understanding these concepts is crucial for solving the problem efficiently. We will delve into these operations step by step to ensure a clear understanding of the solution process.

Matrix Scalar Multiplication

The first step involves scalar multiplication, where we multiply the matrix by the scalar 3. This means we multiply each element inside the matrix by 3. Let's perform this operation on the first matrix:

3(5xy4)=(3∗53∗x3∗y3∗4)=(153x3y12)3\begin{pmatrix} 5 & x \\ y & 4 \end{pmatrix} = \begin{pmatrix} 3*5 & 3*x \\ 3*y & 3*4 \end{pmatrix} = \begin{pmatrix} 15 & 3x \\ 3y & 12 \end{pmatrix}

So, after multiplying by the scalar, our equation now looks like this:

(153x3y12)+(−6−43−y−7)=(98135)\begin{pmatrix} 15 & 3x \\ 3y & 12 \end{pmatrix} + \begin{pmatrix} -6 & -4 \\ 3-y & -7 \end{pmatrix} = \begin{pmatrix} 9 & 8 \\ 13 & 5 \end{pmatrix}

Matrix Addition

Next up, we need to add the two matrices on the left-hand side of the equation. Remember, to add matrices, we simply add the corresponding elements. This gives us:

(15+(−6)3x+(−4)3y+(3−y)12+(−7))=(93x−42y+35)\begin{pmatrix} 15 + (-6) & 3x + (-4) \\ 3y + (3-y) & 12 + (-7) \end{pmatrix} = \begin{pmatrix} 9 & 3x - 4 \\ 2y + 3 & 5 \end{pmatrix}

Now, our equation looks a lot simpler:

(93x−42y+35)=(98135)\begin{pmatrix} 9 & 3x - 4 \\ 2y + 3 & 5 \end{pmatrix} = \begin{pmatrix} 9 & 8 \\ 13 & 5 \end{pmatrix}

Solving for x and y

Now comes the fun part – actually solving for x and y! We can do this by comparing the corresponding elements in the two matrices. Since the matrices are equal, their corresponding elements must be equal as well.

Solving for x

Let's start with x. We can see that the top-right element in the first matrix (3x - 4) must be equal to the top-right element in the second matrix (8). This gives us the equation:

3x−4=83x - 4 = 8

To solve for x, we can add 4 to both sides:

3x=123x = 12

And then divide by 3:

x=4x = 4

So, we've found that x = 4! Awesome!

Solving for y

Now, let's find y. We can compare the bottom-left elements of the matrices. This gives us the equation:

2y+3=132y + 3 = 13

Subtract 3 from both sides:

2y=102y = 10

And then divide by 2:

y=5y = 5

We've got y = 5! Double awesome!

Calculating 2x - y

We're almost there! Now that we know x = 4 and y = 5, we can finally calculate 2x - y. Just plug in the values:

2x−y=2(4)−5=8−5=32x - y = 2(4) - 5 = 8 - 5 = 3

So, the value of 2x - y is 3! We did it!

Answer and Conclusion

Therefore, the correct answer is b. 3. This problem brilliantly showcases how matrix operations can be used to solve for unknowns. By understanding matrix scalar multiplication and matrix addition, we were able to break down the equation and find the values of x and y. Remember guys, the key to success in math is to take things one step at a time and to really understand the underlying concepts. Practice makes perfect, so keep those matrix problems coming!

Key Takeaways

  • Matrix Operations: This problem highlights the importance of understanding basic matrix operations like scalar multiplication and addition.
  • Solving for Unknowns: Matrix equations can be used to solve for unknown variables, much like regular algebraic equations.
  • Step-by-Step Approach: Breaking down a complex problem into smaller, manageable steps makes it easier to solve.
  • Verification: Always double-check your answers by plugging the values back into the original equation.

Why This Problem Matters

Matrix equations are not just abstract mathematical concepts; they have real-world applications in various fields such as computer graphics, physics, and engineering. Understanding how to solve them is a valuable skill. For example, in computer graphics, matrices are used to represent transformations like rotations and scaling. In physics, they can be used to describe systems of linear equations. In engineering, matrices play a crucial role in structural analysis and circuit design. The ability to manipulate and solve matrix equations is therefore essential for professionals in these fields.

Tips for Success

  • Practice Regularly: The more you practice, the more comfortable you will become with matrix operations and equation solving.
  • Understand the Concepts: Don't just memorize formulas; understand the underlying concepts.
  • Break It Down: Divide complex problems into smaller, more manageable steps.
  • Double-Check: Always double-check your work to avoid errors.

By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of matrix problems. So keep practicing, keep learning, and keep exploring the fascinating world of matrices!

I hope this explanation was helpful, guys! Keep practicing, and you'll be matrix masters in no time! ✨