Solve The System Of Equations:${ \begin{array}{l} x = 4y \ x + 4 = 12 \end{array} }$

Mar 13, 2025 by ADMIN 86 views

Solving the System of Equations: A Step-by-Step Guide

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Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.

The System of Equations

The given system of equations is:

\begin{array}{l} x = 4y \\ x + 4 = 12 \end{array}

Step 1: Write Down the Given Equations

The first equation is $x = 4y$ , and the second equation is $x + 4 = 12$ . We will use these equations to solve for the values of $x$ and $y$ .

Step 2: Solve the First Equation for x

The first equation is $x = 4y$ . We can solve for $x$ by multiplying both sides of the equation by $1$ , which gives us:

x = 4y

Step 3: Substitute the Expression for x into the Second Equation

We can substitute the expression for $x$ into the second equation by replacing $x$ with $4y$ . This gives us:

4y + 4 = 12

Step 4: Solve for y

We can solve for $y$ by subtracting $4$ from both sides of the equation, which gives us:

4y = 8

Next, we can divide both sides of the equation by $4$ to get:

y = 2

Step 5: Find the Value of x

Now that we have found the value of $y$ , we can substitute it into the expression for $x$ to find the value of $x$ . We have:

x = 4y

Substituting $y = 2$ into this equation gives us:

x = 4(2)

x = 8

Conclusion

In this article, we have solved a system of two linear equations with two variables. We used the given system of equations as an example and provided a step-by-step guide on how to solve it. We found that the values of $x$ and $y$ are $8$ and $2$ , respectively.

Tips and Tricks

When solving a system of equations, it is often helpful to use substitution or elimination to simplify the equations.
Make sure to check your work by plugging the values of $x$ and $y$ back into the original equations.
If you are having trouble solving a system of equations, try graphing the equations on a coordinate plane to see if they intersect.

Real-World Applications

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

Physics: Solving systems of equations is used to describe the motion of objects in physics.
Engineering: Solving systems of equations is used to design and optimize systems in engineering.
Economics: Solving systems of equations is used to model economic systems and make predictions about the behavior of markets.

Final Thoughts

Solving systems of equations is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve systems of equations and apply the concepts to real-world problems.

Frequently Asked Questions

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve two or more variables.

Q: How do I solve a system of equations?

A: There are several methods for solving systems of equations, including substitution, elimination, and graphing.

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

A: Solving systems of equations has many real-world applications, including physics, engineering, and economics.

Q: How do I check my work when solving a system of equations?

A: To check your work, plug the values of $x$ and $y$ back into the original equations and make sure 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 your work
Not following the steps outlined in the solution
Not using the correct method for solving the system of equations

References

[1] "Solving Systems of Equations" by Math Open Reference
[2] "Systems of Equations" by Khan Academy
[3] "Solving Systems of Equations" by Purplemath

Additional Resources

[1] "Solving Systems of Equations" by Mathway
[2] "Systems of Equations" by IXL
[3] "Solving Systems of Equations" by Wolfram Alpha

Note: The references and additional resources provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.

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Introduction

Solving systems of equations is a fundamental concept in mathematics that has many real-world applications. However, it can be a challenging topic for many students. In this article, we will provide a Q&A guide to help you understand the concept of solving systems of equations and provide you with the tools and resources you need to succeed.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve two or more variables. For example:

\begin{array}{l} x + y = 4 \\ 2x - 3y = -3 \end{array}

Q: How do I solve a system of equations?

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

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

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 your work
Not following the steps outlined in the solution
Not using the correct method for solving the system of equations
Not considering the possibility of multiple solutions or no solution

Q: How do I check my work when solving a system of equations?

A: To check your work, plug the values of $x$ and $y$ back into the original equations and make sure they are true. You can also use a graphing calculator or a computer program to check your work.

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

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

Physics: Solving systems of equations is used to describe the motion of objects in physics.
Engineering: Solving systems of equations is used to design and optimize systems in engineering.
Economics: Solving systems of equations is used to model economic systems and make predictions about the behavior of markets.

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

A: The choice of method depends on the type of system of equations and the variables involved. Here are some general guidelines:

Substitution: Use substitution when one equation is easily solvable for one variable.
Elimination: Use elimination when the coefficients of the variables are easy to add or subtract.
Graphing: Use graphing when the system of equations is simple and the point of intersection is easy to find.

Q: What are some common types of systems of equations?

A: Some common types of systems of equations include:

Linear systems: These are systems of equations where the variables are linear.
Nonlinear systems: These are systems of equations where the variables are nonlinear.
Systems with multiple solutions: These are systems of equations where there are multiple solutions.
Systems with no solution: These are systems of equations where there is no solution.

Q: How do I solve a system of equations with multiple variables?

A: To solve a system of equations with multiple variables, you can use the same methods as before, but you will need to use more variables and equations. Here are some general guidelines:

Substitution: Use substitution to solve for one variable and then substitute that expression into the other equations.
Elimination: Use elimination to add or subtract the equations to eliminate one variable.
Graphing: Use graphing to graph the equations on a coordinate plane and find the point of intersection.

Q: What are some resources for learning more about solving systems of equations?

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

Mathway: This is a online math problem solver that can help you solve systems of equations.
Khan Academy: This is a free online resource that provides video lessons and practice exercises on solving systems of equations.
Purplemath: This is a free online resource that provides lessons and practice exercises on solving systems of equations.
Wolfram Alpha: This is a computer program that can help you solve systems of equations and provide step-by-step solutions.

Conclusion

Solving systems of equations is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article and using the resources provided, you can learn how to solve systems of equations and apply the concepts to real-world problems.

Frequently Asked Questions

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve two or more variables.

Q: How do I solve a system of equations?

A: There are several methods for solving systems of equations, including substitution, elimination, and graphing.

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 your work, not following the steps outlined in the solution, and not using the correct method for solving the system of equations.

Q: How do I check my work when solving a system of equations?

A: To check your work, plug the values of $x$ and $y$ back into the original equations and make sure they are true.

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

A: Solving systems of equations has many real-world applications, including physics, engineering, and economics.

References

[1] "Solving Systems of Equations" by Math Open Reference
[2] "Systems of Equations" by Khan Academy
[3] "Solving Systems of Equations" by Purplemath

Additional Resources

[1] "Solving Systems of Equations" by Mathway
[2] "Systems of Equations" by IXL
[3] "Solving Systems of Equations" by Wolfram Alpha