Solve The System Of Equations:${ \begin{array}{l} 2x + 3y = 4 \ 5x + 6y = 7 \end{array} }$
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Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.
The System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} 2x + 3y = 4 \\ 5x + 6y = 7 \end{array} \}
This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.
Method 1: Substitution Method
One way to solve this system is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.
Let's start by solving the first equation for x:
2x + 3y = 4
Subtracting 3y from both sides gives:
2x = 4 - 3y
Dividing both sides by 2 gives:
x = (4 - 3y) / 2
Now, substitute this expression for x into the second equation:
5x + 6y = 7
Substituting x = (4 - 3y) / 2 into the second equation gives:
5((4 - 3y) / 2) + 6y = 7
Expanding and simplifying the equation gives:
10 - 15y + 12y = 14
Combine like terms:
-3y = 4
Dividing both sides by -3 gives:
y = -4/3
Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the first equation:
2x + 3y = 4
Substituting y = -4/3 into the first equation gives:
2x + 3(-4/3) = 4
Simplifying the equation gives:
2x - 4 = 4
Adding 4 to both sides gives:
2x = 8
Dividing both sides by 2 gives:
x = 4
Therefore, the solution to the system of equations is x = 4 and y = -4/3.
Method 2: Elimination Method
Another way to solve this system is by using the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables.
Let's start by multiplying the first equation by 5 and the second equation by 2:
10x + 15y = 20
10x + 12y = 14
Now, subtract the second equation from the first equation:
(10x + 15y) - (10x + 12y) = 20 - 14
Simplifying the equation gives:
3y = 6
Dividing both sides by 3 gives:
y = 2
Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the first equation:
2x + 3y = 4
Substituting y = 2 into the first equation gives:
2x + 3(2) = 4
Simplifying the equation gives:
2x + 6 = 4
Subtracting 6 from both sides gives:
2x = -2
Dividing both sides by 2 gives:
x = -1
Therefore, the solution to the system of equations is x = -1 and y = 2.
Conclusion
In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. Both methods have given us the same solution: x = 4 and y = -4/3 (using the substitution method) and x = -1 and y = 2 (using the elimination method). This demonstrates that both methods are valid and can be used to solve systems of linear equations.
Real-World Applications
Systems of linear equations have many real-world applications, including:
- Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
- Economics: Systems of linear equations are used to model economic systems, such as the supply and demand of goods and services.
- Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.
Final Thoughts
Solving systems of linear equations is an important skill in mathematics and has many real-world applications. By using the substitution method and the elimination method, we can solve systems of linear equations and find the values of the variables that satisfy all the equations in the system.
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Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.
Q: How do I know which method to use to solve a system of linear equations?
A: The choice of method depends on the specific system of equations and the variables involved. The substitution method is often used when one equation is easily solvable for one variable, while the elimination method is often used when the equations are easily added or subtracted to eliminate one of the variables.
Q: What is the difference between the substitution method and the elimination method?
A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables.
Q: Can I use both methods to solve a system of linear equations?
A: Yes, you can use both methods to solve a system of linear equations. However, it's often more efficient to use one method over the other, depending on the specific system of equations.
Q: What if I have a system of linear equations with more than two variables?
A: If you have a system of linear equations with more than two variables, you can use the same methods as before, but you may need to use additional techniques, such as substitution or elimination, to solve for each variable.
Q: Can I use technology to solve systems of linear equations?
A: Yes, you can use technology, such as graphing calculators or computer software, to solve systems of linear equations. These tools can help you visualize the equations and find the solution.
Q: What if I have a system of linear equations with no solution?
A: If you have a system of linear equations with no solution, it means that the equations are inconsistent and there is no value of the variables that satisfies all the equations.
Q: What if I have a system of linear equations with infinitely many solutions?
A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that satisfy all the equations.
Q: Can I use systems of linear equations to model real-world problems?
A: Yes, you can use systems of linear equations to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
Q: What are some common applications of systems of linear equations?
A: Some common applications of systems of linear equations include:
- Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
- Economics: Systems of linear equations are used to model economic systems, such as the supply and demand of goods and services.
- Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.
Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?
A: You can determine the type of solution by examining the equations and using techniques such as substitution or elimination to solve for the variables.
Q: Can I use systems of linear equations to solve optimization problems?
A: Yes, you can use systems of linear equations to solve optimization problems, such as finding the maximum or minimum value of a function subject to certain constraints.
Q: What are some common mistakes to avoid when solving systems of linear equations?
A: Some common mistakes to avoid when solving systems of linear equations include:
- Not checking for consistency: Make sure the equations are consistent before solving for the variables.
- Not using the correct method: Choose the correct method for the specific system of equations.
- Not checking for infinitely many solutions: Make sure the equations are not dependent before solving for the variables.
Q: Can I use systems of linear equations to solve systems of nonlinear equations?
A: No, you cannot use systems of linear equations to solve systems of nonlinear equations. Nonlinear equations require different techniques, such as numerical methods or algebraic manipulations, to solve.