Solve The System Of Equations:${ \begin{array}{r} 5x - 3y = 4 \ x + Y = 4 \end{array} }$

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple linear equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables, using the method of substitution and elimination.

What are Systems of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a statement that two expressions are equal, and the variables are the unknown values that we need to find. For example, the system of equations:

{ \begin{array}{r} 5x - 3y = 4 \\ x + y = 4 \end{array} \}

is a system of two linear equations with two variables, x and y.

Why is Solving Systems of Linear Equations Important?

Solving systems of linear equations is important in many real-world applications, such as:

  • Physics and Engineering: Systems of linear equations are used to model physical systems, such as the motion of objects, electrical circuits, and mechanical systems.
  • Economics: Systems of linear equations are used to model economic systems, such as supply and demand, and to make predictions about the behavior of economic variables.
  • Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.

Methods for Solving Systems of Linear Equations

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

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

Solving the System of Equations using Substitution Method

To solve the system of equations using 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 = 4

We can solve this equation for y by subtracting x from both sides:

y = 4 - x

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

The first equation is:

5x - 3y = 4

We can substitute the expression for y into this equation by replacing y with 4 - x:

5x - 3(4 - x) = 4

Step 3: Simplify the Equation

We can simplify the equation by distributing the -3 to the terms inside the parentheses:

5x - 12 + 3x = 4

Combine like terms:

8x - 12 = 4

Add 12 to both sides:

8x = 16

Divide both sides by 8:

x = 2

Step 4: Find the Value of y

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. We will use the second equation:

x + y = 4

Substitute x = 2:

2 + y = 4

Subtract 2 from both sides:

y = 2

Solving the System of Equations using Elimination Method

To solve the system of equations using the elimination method, we will add the two equations to eliminate one variable.

Step 1: Add the Two Equations

The two equations are:

5x - 3y = 4

x + y = 4

We can add these equations by adding the corresponding terms:

(5x + x) + (-3y + y) = 4 + 4

Combine like terms:

6x - 2y = 8

Step 2: Simplify the Equation

We can simplify the equation by dividing both sides by 2:

3x - y = 4

Step 3: Solve for y

We can solve for y by subtracting 3x from both sides:

-y = 4 - 3x

Multiply both sides by -1:

y = 3x - 4

Step 4: Find the Value of x

Now that we have found the value of y, we can substitute it into one of the original equations to find the value of x. We will use the first equation:

5x - 3y = 4

Substitute y = 3x - 4:

5x - 3(3x - 4) = 4

Distribute the -3:

5x - 9x + 12 = 4

Combine like terms:

-4x + 12 = 4

Subtract 12 from both sides:

-4x = -8

Divide both sides by -4:

x = 2

Step 5: Find the Value of y

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. We will use the second equation:

x + y = 4

Substitute x = 2:

2 + y = 4

Subtract 2 from both sides:

y = 2

Conclusion

Solving systems of linear equations is an important concept in mathematics, particularly in algebra and geometry. In this article, we have discussed the methods of substitution and elimination for solving systems of linear equations, and we have applied these methods to solve a system of two linear equations with two variables. We have found that the values of x and y are x = 2 and y = 2, respectively.

Future Work

In the future, we can explore other methods for solving systems of linear equations, such as the graphical method and the matrix method. We can also apply these methods to solve systems of linear equations with more variables.

References

  • [1] "Systems of Linear Equations" by Khan Academy
  • [2] "Solving Systems of Linear Equations" by Math Open Reference
  • [3] "Systems of Linear Equations" by Wolfram MathWorld

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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 two or more variables. Each equation is a statement that two expressions are equal, and the variables are the unknown values that we need to find.

Q: What are the methods for solving systems of linear equations?

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

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

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

A: The choice of method depends on the type of system and the variables involved. For example, if the system has two variables and two equations, the substitution method may be the most straightforward approach. If the system has more variables or equations, the elimination method may be more efficient.

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 variable.

Q: How do I solve a system of linear equations using the substitution method?

A: To solve a system of linear equations using the substitution method, follow these steps:

  1. Solve one equation for one variable.
  2. Substitute that expression into the other equation.
  3. Simplify the resulting equation.
  4. Solve for the remaining variable.
  5. Substitute the value of the remaining variable back into one of the original equations to find the value of the other variable.

Q: How do I solve a system of linear equations using the elimination method?

A: To solve a system of linear equations using the elimination method, follow these steps:

  1. Add or subtract the equations to eliminate one variable.
  2. Simplify the resulting equation.
  3. Solve for the remaining variable.
  4. Substitute the value of the remaining variable back into one of the original equations to find the value of the other variable.

Q: What is the graphical method for solving systems of linear equations?

A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful for visualizing the relationships between the variables and for finding approximate solutions.

Q: How do I graph a system of linear equations?

A: To graph a system of linear equations, follow these steps:

  1. Graph each equation on a coordinate plane.
  2. Find the point of intersection between the two graphs.
  3. The point of intersection represents the solution to the system.

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 the solution: Make sure to check the solution by substituting it back into the original equations.
  • Not following the order of operations: Make sure to follow the order of operations when simplifying the equations.
  • Not using the correct method: Make sure to use the correct method for the type of system.

Q: How do I check the solution to a system of linear equations?

A: To check the solution to a system of linear equations, follow these steps:

  1. Substitute the solution back into the original equations.
  2. Simplify the resulting equations.
  3. If the solution satisfies both equations, it is the correct solution.

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 and Engineering: Systems of linear equations are used to model physical systems, such as the motion of objects, electrical circuits, and mechanical systems.
  • Economics: Systems of linear equations are used to model economic systems, such as supply and demand, and to make predictions about the behavior of economic variables.
  • Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.

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 and find the point of intersection.
  • Computer algebra systems: Computer algebra systems, such as Mathematica and Maple, can be used to solve systems of linear equations and to graph the equations.
  • Online tools: Online tools, such as Wolfram Alpha and Mathway, can be used to solve systems of linear equations and to graph the equations.

Q: What are some advanced topics in solving systems of linear equations?

A: Some advanced topics in solving systems of linear equations include:

  • Matrix methods: Matrix methods involve using matrices to represent the system of equations and to solve for the variables.
  • Determinants: Determinants are used to find the solution to a system of linear equations.
  • Inverse matrices: Inverse matrices are used to solve systems of linear equations.

Conclusion

Solving systems of linear equations is an important concept in mathematics, particularly in algebra and geometry. In this article, we have discussed the methods of substitution and elimination for solving systems of linear equations, and we have answered some frequently asked questions about solving systems of linear equations. We have also explored some advanced topics in solving systems of linear equations, including matrix methods, determinants, and inverse matrices.