Solve The Following System Of Equations:${ \begin{array}{l} 5x + Y = 4 \ x - Y = 2 \end{array} }$

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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.

What is a System of Linear Equations?

A system of linear equations is a set of equations in which each equation is a linear equation. A linear equation is an equation in which the highest power of the variable(s) is 1. For example, the equation 2x + 3y = 5 is a linear equation, while the equation x^2 + 2y = 3 is not a linear equation.

Types of Systems of Linear Equations

There are three types of systems of linear equations:

  • Consistent system: A consistent system is a system in which there is at least one solution. In other words, there is at least one set of values for the variables that satisfies all the equations in the system.
  • Inconsistent system: An inconsistent system is a system in which there is no solution. In other words, there is no set of values for the variables that satisfies all the equations in the system.
  • Dependent system: A dependent system is a system in which there are infinitely many solutions. In other words, there are many sets of values for the variables that satisfy all the equations in the system.

Solving a System of Linear Equations

There are several methods for solving a system of linear equations, including:

  • Substitution method: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination method: The elimination method involves adding or subtracting the equations in the system to eliminate one of the variables.
  • Graphical method: The graphical method involves graphing the equations in the system on a coordinate plane and finding the point of intersection.

Solving the Given System of Equations

The given system of equations is:

{ \begin{array}{l} 5x + y = 4 \\ x - y = 2 \end{array} \}

To solve this system of equations, we can use the substitution method. We will solve the second equation for y and then substitute that expression into the first equation.

Step 1: Solve the Second Equation for y

The second equation is x - y = 2. We can solve this equation for y by adding y to both sides and then subtracting x from both sides.

y = x - 2

Step 2: Substitute the Expression for y into the First Equation

The first equation is 5x + y = 4. We can substitute the expression for y that we found in Step 1 into this equation.

5x + (x - 2) = 4

Step 3: Simplify the Equation

We can simplify the equation by combining like terms.

5x + x - 2 = 4

Combine like terms:

6x - 2 = 4

Add 2 to both sides:

6x = 6

Divide both sides by 6:

x = 1

Step 4: Find the Value of y

Now that we have found the value of x, we can find the value of y by substituting x into one of the original equations. We will use the second equation.

x - y = 2

Substitute x = 1:

1 - y = 2

Add y to both sides:

1 = 2 + y

Subtract 2 from both sides:

-1 = y

Conclusion

In this article, we solved a system of two linear equations with two variables using the substitution method. We found that the solution to the system is x = 1 and y = -1.

Example Use Cases

Solving systems of linear equations has many practical applications in mathematics and science. Here are a few examples:

  • Physics: In physics, systems of linear equations are used to model the motion of objects. For example, the equations of motion for an object under the influence of gravity can be represented as a system of linear equations.
  • Engineering: In engineering, systems of linear equations are used to design and optimize systems. For example, the equations for the stress and strain on a beam can be represented as a system of linear equations.
  • Economics: In economics, systems of linear equations are used to model economic systems. For example, the equations for the supply and demand of a good can be represented as a system of linear equations.

Conclusion

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: What are the different types of systems of linear equations?

A: There are three types of systems of linear equations:

  • Consistent system: A consistent system is a system in which there is at least one solution. In other words, there is at least one set of values for the variables that satisfies all the equations in the system.
  • Inconsistent system: An inconsistent system is a system in which there is no solution. In other words, there is no set of values for the variables that satisfies all the equations in the system.
  • Dependent system: A dependent system is a system in which there are infinitely many solutions. In other words, there are many sets of values for the variables that satisfy all the equations in the system.

Q: What are the different methods for solving a system of linear equations?

A: There are several methods for solving a system of linear equations, including:

  • Substitution method: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination method: The elimination method involves adding or subtracting the equations in the system to eliminate one of the variables.
  • Graphical method: The graphical method involves graphing the equations in the system on a coordinate plane and finding the point of intersection.

Q: How do I choose the best method for solving a system of linear equations?

A: The best method for solving a system of linear equations depends on the specific system and the variables involved. If the system has two variables, the substitution method or elimination method may be the best choice. If the system has more than two variables, the graphical method may be more effective.

Q: What are some common mistakes to avoid when solving a system of linear equations?

A: Some common mistakes to avoid when solving a system of linear equations include:

  • Not checking for consistency: Make sure to check if the system is consistent before solving it.
  • Not using the correct method: Choose the best method for the specific system and variables involved.
  • Not checking for infinitely many solutions: Make sure to check if the system has infinitely many solutions before solving it.

Q: How do I check if a system of linear equations has infinitely many solutions?

A: To check if a system of linear equations has infinitely many solutions, you can use the following steps:

  • Check if the equations are dependent: If the equations are dependent, then the system has infinitely many solutions.
  • Check if the equations are inconsistent: If the equations are inconsistent, then the system has no solution.
  • Check if the equations are consistent but have infinitely many solutions: If the equations are consistent but have infinitely many solutions, then the system has infinitely many solutions.

Q: What are some real-world applications of solving systems of linear equations?

A: Solving systems of linear equations has many real-world applications, including:

  • Physics: In physics, systems of linear equations are used to model the motion of objects.
  • Engineering: In engineering, systems of linear equations are used to design and optimize systems.
  • Economics: In economics, systems of linear equations are used to model economic systems.

Q: How do I use technology to solve systems of linear equations?

A: There are many software programs and online tools that can be used to solve systems of linear equations, including:

  • Graphing calculators: Graphing calculators can be used to graph the equations in the system and find the point of intersection.
  • Computer algebra systems: Computer algebra systems can be used to solve systems of linear equations using the substitution method, elimination method, or graphical method.
  • Online calculators: Online calculators can be used to solve systems of linear equations using the substitution method, elimination method, or graphical method.

Conclusion

In conclusion, solving systems of linear equations is an important topic in mathematics and has many practical applications in science and engineering. By understanding the different types of systems of linear equations, the different methods for solving them, and the common mistakes to avoid, you can become proficient in solving systems of linear equations and apply this knowledge to real-world problems.