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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solution to the system of equations.

The System of Equations

The system of equations we will be solving is:

3x+4y=14x+5y=0\begin{array}{l} 3x + 4y = 1 \\ 4x + 5y = 0 \end{array}

Step 1: Multiply the Two Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the two equations by necessary multiples such that the coefficients of either x or y in both equations are the same. Let's multiply the first equation by 5 and the second equation by 4.

15x+20y=516x+20y=0\begin{array}{l} 15x + 20y = 5 \\ 16x + 20y = 0 \end{array}

Step 2: Subtract the Two Equations

Now, we can subtract the second equation from the first equation to eliminate the variable y.

(15x+20y)−(16x+20y)=5−0−x=5\begin{array}{l} (15x + 20y) - (16x + 20y) = 5 - 0 \\ - x = 5 \end{array}

Step 3: Solve for x

Now, we can solve for x by multiplying both sides of the equation by -1.

−x=5x=−5\begin{array}{l} -x = 5 \\ x = -5 \end{array}

Step 4: Substitute x into One of the Original Equations

Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's substitute x into the first equation.

3(−5)+4y=1−15+4y=1\begin{array}{l} 3(-5) + 4y = 1 \\ -15 + 4y = 1 \end{array}

Step 5: Solve for y

Now, we can solve for y by adding 15 to both sides of the equation and then dividing both sides by 4.

−15+4y=14y=16y=4\begin{array}{l} -15 + 4y = 1 \\ 4y = 16 \\ y = 4 \end{array}

The Solution to the System of Equations

The solution to the system of equations is x = -5 and y = 4.

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have shown that the solution to the system of equations is x = -5 and y = 4. This method can be used to solve systems of equations with two variables.

The solution to the system of equations is (-5, 4).

Why is it Important to Solve Systems of Equations?

Solving systems of equations is an important concept in mathematics and has many real-world applications. It is used in fields such as physics, engineering, economics, and computer science. For example, in physics, systems of equations are used to describe the motion of objects and to solve problems involving forces and energies. In engineering, systems of equations are used to design and optimize systems such as bridges, buildings, and electronic circuits.

How to Solve Systems of Equations?

There are several methods to solve systems of equations, including:

  • Substitution Method: This method involves substituting one equation into the other equation to solve for one variable.
  • Elimination Method: This method involves adding or subtracting the two equations to eliminate one variable.
  • Graphical Method: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.
  • Matrix Method: This method involves using matrices to solve systems of equations.

Real-World Applications of Systems of Equations

Systems of equations have many real-world applications. Some examples include:

  • Physics: Systems of equations are used to describe the motion of objects and to solve problems involving forces and energies.
  • Engineering: Systems of equations are used to design and optimize systems such as bridges, buildings, and electronic circuits.
  • Economics: Systems of equations are used to model economic systems and to solve problems involving supply and demand.
  • Computer Science: Systems of equations are used to solve problems involving algorithms and data structures.

Conclusion

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: How do I know if a system of equations has a solution?

A: A system of equations has a solution if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system of equations does not have a solution.

Q: What are the different methods to solve systems of equations?

A: There are several methods to solve systems of equations, including:

  • Substitution Method: This method involves substituting one equation into the other equation to solve for one variable.
  • Elimination Method: This method involves adding or subtracting the two equations to eliminate one variable.
  • Graphical Method: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.
  • Matrix Method: This method involves using matrices to solve systems of equations.

Q: How do I choose the method to solve a system of equations?

A: The method to choose depends on the type of equations and the variables involved. For example, if the equations are linear, then the substitution or elimination method may be used. If the equations are non-linear, then the graphical or matrix method may be used.

Q: What is the difference between a system of linear equations and a system of non-linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. A system of non-linear equations is a set of two or more non-linear equations that are solved simultaneously to find the values of the variables.

Q: How do I solve a system of non-linear equations?

A: To solve a system of non-linear equations, you can use the graphical or matrix method. The graphical method involves graphing the two equations on a coordinate plane and finding the point of intersection. The matrix method involves using matrices to solve the system of equations.

Q: What is the significance of systems of equations in real-world applications?

A: Systems of equations have many real-world applications, including physics, engineering, economics, and computer science. They are used to model and solve problems involving forces and energies, design and optimize systems, model economic systems, and solve problems involving algorithms and data structures.

Q: How do I check if a solution to a system of equations is correct?

A: To check if a solution to a system of equations is correct, you can substitute the values of the variables into both equations and check if they are true.

Q: What are some common mistakes to avoid when solving systems of equations?

A: Some common mistakes to avoid when solving systems of equations include:

  • Not checking if the equations are consistent: Make sure that the two equations are consistent before solving the system of equations.
  • Not choosing the correct method: Choose the correct method to solve the system of equations based on the type of equations and the variables involved.
  • Not checking if the solution is correct: Check if the solution to the system of equations is correct by substituting the values of the variables into both equations.

Conclusion

In conclusion, systems of equations are an important concept in mathematics and have many real-world applications. They are used to model and solve problems involving forces and energies, design and optimize systems, model economic systems, and solve problems involving algorithms and data structures. By understanding the different methods to solve systems of equations and avoiding common mistakes, you can solve systems of equations effectively and efficiently.