Solve Simultaneously For $x$ And $y$:1. $4x + Y = 7$2. $3x^2 + 2xy = Y^2$
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Introduction
Solving systems of nonlinear equations can be a challenging task in mathematics. These equations involve variables raised to powers other than one, making it difficult to isolate the variables and solve for them. In this article, we will explore how to solve simultaneously for x and y in a system of nonlinear equations. We will use the given system of equations as an example and provide a step-by-step solution.
The System of Equations
The given system of equations is:
- 4x + y = 7
- 3x^2 + 2xy = y^2
We are required to solve simultaneously for x and y in this system of equations.
Step 1: Rearrange the First Equation
The first equation is already in a simple form, but we can rearrange it to isolate y:
y = 7 - 4x
Step 2: Substitute the Expression for y into the Second Equation
Now, we will substitute the expression for y from the first equation into the second equation:
3x^2 + 2x(7 - 4x) = (7 - 4x)^2
Step 3: Expand and Simplify the Equation
We will expand and simplify the equation:
3x^2 + 14x - 8x^2 = 49 - 56x + 16x^2
Step 4: Combine Like Terms
We will combine like terms:
-5x^2 + 14x + 56x = 49
Step 5: Simplify the Equation
We will simplify the equation:
-5x^2 + 70x = 49
Step 6: Rearrange the Equation
We will rearrange the equation to make it a quadratic equation:
5x^2 - 70x + 49 = 0
Step 7: Solve the Quadratic Equation
We will solve the quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 5, b = -70, and c = 49.
x = (70 ± √((-70)^2 - 4(5)(49))) / (2(5))
Step 8: Simplify the Expression
We will simplify the expression:
x = (70 ± √(4900 - 980)) / 10
x = (70 ± √3920) / 10
x = (70 ± 62.6) / 10
Step 9: Find the Values of x
We will find the values of x:
x = (70 + 62.6) / 10
x = 132.6 / 10
x = 13.26
x = (70 - 62.6) / 10
x = 7.4 / 10
x = 0.74
Step 10: Find the Values of y
We will find the values of y by substituting the values of x into the expression for y:
y = 7 - 4x
For x = 13.26:
y = 7 - 4(13.26)
y = 7 - 52.04
y = -45.04
For x = 0.74:
y = 7 - 4(0.74)
y = 7 - 2.96
y = 4.04
Conclusion
In this article, we have solved simultaneously for x and y in a system of nonlinear equations. We have used the given system of equations as an example and provided a step-by-step solution. The final values of x and y are x = 13.26 and y = -45.04, and x = 0.74 and y = 4.04.
Final Answer
The final answer is:
x = 13.26, y = -45.04
x = 0.74, y = 4.04
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Introduction
In our previous article, we explored how to solve simultaneously for x and y in a system of nonlinear equations. We used the given system of equations as an example and provided a step-by-step solution. In this article, we will answer some frequently asked questions related to solving systems of nonlinear equations.
Q: What is a system of nonlinear equations?
A system of nonlinear equations is a set of equations that involve variables raised to powers other than one. These equations are nonlinear because they do not have a linear relationship between the variables.
Q: How do I know if a system of equations is nonlinear?
You can determine if a system of equations is nonlinear by looking at the highest power of each variable in the equations. If any of the variables are raised to a power other than one, the system is nonlinear.
Q: What are some common methods for solving systems of nonlinear equations?
Some common methods for solving systems of nonlinear equations include:
- Substitution method: This involves substituting one equation into another to eliminate one of the variables.
- Elimination method: This involves adding or subtracting equations to eliminate one of the variables.
- Graphical method: This involves graphing the equations on a coordinate plane and finding the intersection points.
- Numerical method: This involves using numerical methods such as the Newton-Raphson method to find the solution.
Q: What is the substitution method?
The substitution method involves substituting one equation into another to eliminate one of the variables. This method is useful when one of the equations is linear and the other is nonlinear.
Q: What is the elimination method?
The elimination method involves adding or subtracting equations to eliminate one of the variables. This method is useful when the coefficients of the variables are the same in both equations.
Q: What is the graphical method?
The graphical method involves graphing the equations on a coordinate plane and finding the intersection points. This method is useful when the system of equations has a small number of solutions.
Q: What is the numerical method?
The numerical method involves using numerical methods such as the Newton-Raphson method to find the solution. This method is useful when the system of equations has a large number of solutions.
Q: How do I choose the best method for solving a system of nonlinear equations?
The best method for solving a system of nonlinear equations depends on the specific system and the number of solutions. You should choose the method that is most suitable for the system and the number of solutions.
Q: What are some common mistakes to avoid when solving systems of nonlinear equations?
Some common mistakes to avoid when solving systems of nonlinear equations include:
- Not checking the validity of the solutions
- Not checking for extraneous solutions
- Not using the correct method for the system
- Not checking for multiple solutions
Conclusion
In this article, we have answered some frequently asked questions related to solving systems of nonlinear equations. We have discussed the different methods for solving systems of nonlinear equations and provided some tips for choosing the best method. We have also discussed some common mistakes to avoid when solving systems of nonlinear equations.
Final Answer
The final answer is:
- The substitution method is useful when one of the equations is linear and the other is nonlinear.
- The elimination method is useful when the coefficients of the variables are the same in both equations.
- The graphical method is useful when the system of equations has a small number of solutions.
- The numerical method is useful when the system of equations has a large number of solutions.
- The best method for solving a system of nonlinear equations depends on the specific system and the number of solutions.
- Some common mistakes to avoid when solving systems of nonlinear equations include not checking the validity of the solutions, not checking for extraneous solutions, not using the correct method for the system, and not checking for multiple solutions.