To solve the system represented by the given augmented matrix, we first need to interpret the matrix and then perform row operations to find the solutions.
The augmented matrix is:
This corresponds to the following system of equations:
Step 1: Row Reduction
We will perform row operations to simplify this system. We can start by eliminating from the second equation.
To eliminate from the second equation, we can multiply the first equation by 2 and subtract it from the second equation:
This simplifies to:
Which gives us:
Now, we can rewrite the augmented matrix as:
Step 2: Solve for One Variable
From the second equation, we can express in terms of :
Step 3: Parameterize the Free Variable
Let (where is a free variable). Then we can express in terms of :
Now, substituting and back into the first equation to find :
Solving for :
Final Parameterized Solution
Now we can summarize the parameterized set of solutions:
Thus, the parameterized set of solutions for the system is:
Where is a free parameter.
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The Deep Dive
To solve the augmented matrix given by
first, we can represent this as a system of linear equations:
Next, we can express one variable in terms of the others. Let’s denote as the free variable . From the first equation, we can solve for :
Now, substituting into the second equation gives us:
Simplifying this will allow us to express in terms of .
After a bit of calculation, we get:
Now, substituting back into our expression for ,
you can simplify this expression.
The parameterized solution set will then look like:
Finally, don’t forget that can take any real number value!
Did you know that matrices and their applications extend far beyond just solving systems of linear equations? In fields like computer graphics, they’re crucial for transforming images and models. By manipulating matrices, you can rotate, scale, and translate visual data in stunning ways—an essential technique in game design and animation!
If you find joy in practicing matrices, consider diving into linear programming! This mathematical method is used in business for optimizing resource allocation, like maximizing profits or minimizing costs. It elegantly combines systems of equations with real-world scenarios, making it a treasure trove for practical applications in economics and operations research!