Solve The Following System Of Equations:1. $5x + 6x = 13$2. $y = -x + 4$3. 5 X + 6 Y = 13 5x + 6y = 13 5 X + 6 Y = 13

Mar 12, 2025 by ADMIN 118 views

**Solving a System of Linear Equations**

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 solve a system of three linear equations with two variables.

The System of Equations

The system of equations we will solve is:

$5x + 6x = 13$
$y = -x + 4$
$5x + 6y = 13$

Step 1: Simplify the First Equation

The first equation is $5x + 6x = 13$ . We can simplify this equation by combining like terms. The like terms in this equation are $5x$ and $6x$ , which can be combined to form $11x$ . Therefore, the simplified equation is:

$11x = 13$

Step 2: Solve for x

To solve for $x$ , we need to isolate $x$ on one side of the equation. We can do this by dividing both sides of the equation by $11$ . This gives us:

$x = \frac{13}{11}$

Step 3: Substitute x into the Second Equation

The second equation is $y = -x + 4$ . We can substitute the value of $x$ we found in Step 2 into this equation. This gives us:

$y = -\frac{13}{11} + 4$

Step 4: Simplify the Second Equation

To simplify the second equation, we need to get rid of the fraction. We can do this by multiplying both sides of the equation by $11$ . This gives us:

$11y = -13 + 44$

Step 5: Solve for y

To solve for $y$ , we need to isolate $y$ on one side of the equation. We can do this by dividing both sides of the equation by $11$ . This gives us:

$y = \frac{31}{11}$

Step 6: Substitute x and y into the Third Equation

The third equation is $5x + 6y = 13$ . We can substitute the values of $x$ and $y$ we found in Steps 2 and 5 into this equation. This gives us:

$5\left(\frac{13}{11}\right) + 6\left(\frac{31}{11}\right) = 13$

Step 7: Simplify the Third Equation

To simplify the third equation, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by $11$ . This gives us:

$65 + 186 = 143$

Step 8: Check the Solution

To check the solution, we need to make sure that the values of $x$ and $y$ we found satisfy all three equations in the system. We can do this by plugging the values of $x$ and $y$ into each equation and checking if the equation is true.

Conclusion

In this article, we solved a system of three linear equations with two variables. We used the substitution method to solve for $x$ and $y$ , and then checked the solution to make sure it satisfied all three equations in the system. The values of $x$ and $y$ we found are $x = \frac{13}{11}$ and $y = \frac{31}{11}$ .

Example Use Cases

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

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 written 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 written as a system of linear equations.
Computer Science: In computer science, systems of linear equations are used to solve problems in computer graphics and game development. For example, the equations for the position and velocity of an object in a game can be written as a system of linear equations.

Tips and Tricks

Here are a few tips and tricks for solving systems of linear equations:

Use the substitution method: The substitution method is a powerful tool for solving systems of linear equations. It involves substituting the value of one variable into the other equations in the system.
Use the elimination method: The elimination method is another powerful tool for solving systems of linear equations. It involves eliminating one variable from the equations in the system.
Check the solution: It's always a good idea to check the solution to a system of linear equations to make sure it satisfies all the equations in the system.

Conclusion

Introduction

In our previous article, we solved a system of three linear equations with two variables. We used the substitution method to solve for $x$ and $y$ , and then checked the solution to make sure it satisfied all three equations in the system. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

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 if a system of linear equations has a solution?

A system of linear equations has a solution if and only if the equations are consistent. In other words, if the equations are true for the same values of the variables, then the system has a solution.

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

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

Substitution method: This method involves substituting the value of one variable into the other equations in the system.
Elimination method: This method involves eliminating one variable from the equations in the system.
Graphical method: This method involves graphing the equations 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?

The best method for solving a system of linear equations depends on the specific equations and the variables involved. Here are some general guidelines:

Use the substitution method: If one of the equations is already solved for one of the variables, use the substitution method.
Use the elimination method: If the coefficients of one of the variables are the same in both equations, use the elimination method.
Use the graphical method: If the equations are simple and the variables are easy to graph, use the graphical method.

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

Here are some common mistakes to avoid when solving systems of linear equations:

Not checking the solution: Always check the solution to make sure it satisfies all the equations in the system.
Not using the correct method: Choose the best method for solving the system, and use it correctly.
Not simplifying the equations: Simplify the equations as much as possible to make them easier to solve.

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

To check the solution to a system of linear equations, plug the values of the variables into each equation and check if the equation is true. If the equation is true, then the solution is correct.

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

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.
Computer Science: In computer science, systems of linear equations are used to solve problems in computer graphics and game development.

Conclusion

In conclusion, solving systems of linear equations is an important skill in mathematics and science. By understanding the different methods for solving systems of linear equations and avoiding common mistakes, we can solve systems of linear equations with ease. Whether you are a student, a teacher, or a professional, solving systems of linear equations is an essential skill that will serve you well in your career.

Additional Resources

Here are some additional resources for learning more about solving systems of linear equations:

Textbooks: There are many textbooks available on solving systems of linear equations, including "Linear Algebra and Its Applications" by Gilbert Strang and "Introduction to Linear Algebra" by Gilbert Strang.
Online Resources: There are many online resources available for learning about solving systems of linear equations, including Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
Practice Problems: Practice problems are an essential part of learning about solving systems of linear equations. You can find practice problems online or in textbooks.

Final Thoughts

Solving systems of linear equations is a fundamental skill in mathematics and science. By understanding the different methods for solving systems of linear equations and avoiding common mistakes, we can solve systems of linear equations with ease. Whether you are a student, a teacher, or a professional, solving systems of linear equations is an essential skill that will serve you well in your career.