3x3-Systems: Problem 3 5xy+5z=94x+5y5z=-28x+3y+2z=-7Find The Unique Solution To This System Of Equations.
Solving systems of equations is a fundamental aspect of algebra and mathematics in general. When dealing with a 3x3 system, the goal is to find the unique set of values for the variables that satisfy all three equations simultaneously. In this guide, we will thoroughly analyze and solve the problem: "3x3-Systems: Problem 3 5xy+5z=94x+5y5z=-28x+3y+2z=-7," aiming to find the unique solution for this system of equations.
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Understanding the System of Equations
Before diving into the solution process, it is essential to understand the structure and components of the given system.
Presented System of Equations
The system appears as follows:
- 5xy + 5z = 94
- 5z = -28
- 4x + 3y + 2z = -7
At first glance, it seems that the first equation involves a product of variables (xy), which indicates that this is a nonlinear system due to the xy term. The second equation is linear in z, and the third is a standard linear equation involving x, y, and z.
Clarifying the Equations
Given the notation, it appears the original problem may have some typographical errors or formatting issues. Based on standard conventions and the context, the system is likely:
- Equation 1: 5xy + 5z = 94
- Equation 2: 5z = -28
- Equation 3: 4x + 3y + 2z = -7
This interpretation makes sense because:
- Equation 2 is straightforward: 5z = -28, which allows us to directly find z.
- Equations 1 and 3 involve x and y, with the first involving xy (product of x and y), indicating a nonlinear component.
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Step-by-Step Solution Strategy
To find the unique solution for the system, the approach involves:
- Solve for z directly from the simpler equation.
- Substitute the value of z into the other equations.
- Handle the nonlinear term involving xy in the first equation.
- Solve the resulting system of equations for x and y.
Let's proceed with this plan.
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Step 1: Solving for z
From equation 2:
5z = -28
Dividing both sides by 5:
z = -28 / 5 = -5.6
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Step 2: Substituting z into the other equations
With z known, substitute into equations 1 and 3.
Equation 1:
5xy + 5z = 94
Substitute z = -5.6:
5xy + 5 (-5.6) = 94
Calculate:
5xy - 28 = 94
Add 28 to both sides:
5xy = 94 + 28 = 122
Divide both sides by 5:
xy = 122 / 5 = 24.4
Equation 3:
4x + 3y + 2z = -7
Substitute z = -5.6:
4x + 3y + 2 (-5.6) = -7
Calculate:
4x + 3y - 11.2 = -7
Add 11.2 to both sides:
4x + 3y = 4.2
Now, the system reduces to:
- xy = 24.4
- 4x + 3y = 4.2
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Step 3: Solving for x and y
We now have a nonlinear equation (xy = 24.4) and a linear equation (4x + 3y = 4.2). To solve this, express y in terms of x using the linear equation, then substitute into xy.
Express y in terms of x:
From:
4x + 3y = 4.2
Solve for y:
3y = 4.2 - 4x
\[
y = \frac{4.2 - 4x}{3}
\]
Substitute into xy = 24.4:
\[
x y = 24.4
\]
\[
x \left( \frac{4.2 - 4x}{3} \right) = 24.4
\]
Multiply both sides by 3 to clear denominator:
\[
x (4.2 - 4x) = 73.2
\]
Distribute x:
\[
4.2x - 4x^2 = 73.2
\]
Rewrite as a quadratic in x:
\[
-4x^2 + 4.2x - 73.2 = 0
\]
Divide through by -4 to simplify:
\[
x^2 - 1.05x + 18.3 = 0
\]
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Step 4: Solving the Quadratic Equation
Quadratic:
\[
x^2 - 1.05x + 18.3 = 0
\]
Calculate discriminant:
\[
D = (−1.05)^2 - 4 1 18.3 = 1.1025 - 73.2 = -72.0975
\]
Since the discriminant is negative, this indicates no real solutions for x.
Implication:
Because the quadratic has no real solutions, the system has no real solution that satisfies all the equations simultaneously.
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Conclusion: Nature of the Solution
Based on the calculations, the system:
- Has a unique solution for z, which is z = -5.6.
- Leads to a quadratic in x with a negative discriminant, indicating no real solutions for x and y.
Therefore, the system has no real solutions.
However, if complex solutions are acceptable, the quadratic can be solved in the complex domain:
\[
x = \frac{1.05 \pm \sqrt{-72.0975}}{2}
\]
Expressed in complex form:
\[
x = \frac{1.05 \pm i \sqrt{72.0975}}{2}
\]
Correspondingly, y can be computed from y = (4.2 - 4x) / 3.
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Summary and Key Takeaways
- The initial step was to identify and interpret the system correctly.
- Directly solving for z was straightforward, yielding z = -5.6.
- Substituting z into the other equations reduced the problem to solving a quadratic in x.
- The quadratic had a negative discriminant, indicating no real solutions.
- The system, therefore, has no real solutions but complex solutions exist.
Additional Tips for Solving Nonlinear 3x3 Systems
- Always verify the system's structure and check for typographical errors.
- Isolate variables step-by-step, starting with the simplest equations.
- Be prepared to handle nonlinear equations involving products of variables.
- Use substitution to reduce the system to a quadratic or simpler form.
- Carefully compute the discriminant to determine the nature of solutions.
- When no real solutions exist, consider complex solutions if applicable.
Final Thoughts
Solving 3x3 systems, especially those involving nonlinear components like xy, requires careful algebraic manipulation and sometimes advanced techniques. In this particular problem, the key was recognizing the nonlinear term and systematically reducing the system to a quadratic. The negative discriminant indicated that the system has no solutions in the real number domain, but complex solutions are possible.
Understanding these processes enhances problem-solving skills and deepens comprehension of algebraic systems, which are essential for higher mathematics, engineering, physics, and related fields.
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If you have further questions on solving complex systems of equations or need assistance with similar problems, feel free to ask!
