Solve The System By Substitution.$\begin{aligned} -5x + 3y &= 2 \\ y &= 2x \end{aligned}$

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Introduction

Solving systems of equations is a fundamental concept in mathematics, and there are several methods to approach this problem. In this article, we will focus on solving systems of equations by substitution. This method involves substituting one equation into another to solve for the variables. We will use a system of linear equations as an example to demonstrate this method.

The System of Equations

The system of equations we will be working with is:

−5x+3y=2y=2x\begin{aligned} -5x + 3y &= 2 \\ y &= 2x \end{aligned}

Step 1: Write Down the Equations

The first step is to write down the equations. In this case, we have two equations:

  1. −5x+3y=2-5x + 3y = 2
  2. y=2xy = 2x

Step 2: Substitute the Second Equation into the First Equation

Now, we will substitute the second equation into the first equation. We will replace yy with 2x2x in the first equation:

−5x+3(2x)=2-5x + 3(2x) = 2

Step 3: Simplify the Equation

Next, we will simplify the equation by distributing the 3 to the terms inside the parentheses:

−5x+6x=2-5x + 6x = 2

Step 4: Combine Like Terms

Now, we will combine like terms by adding or subtracting the coefficients of the variables:

x=2x = 2

Step 5: Find the Value of y

Now that we have found the value of xx, we can substitute it into the second equation to find the value of yy:

y=2xy = 2x y=2(2)y = 2(2) y=4y = 4

Conclusion

In this article, we have demonstrated how to solve a system of equations by substitution. We used a system of linear equations as an example and followed the steps to find the values of the variables. This method is useful when one of the equations is already solved for one of the variables.

Real-World Applications

Solving systems of equations by substitution has many real-world applications. For example, in physics, we can use this method to solve problems involving motion and forces. In economics, we can use this method to solve problems involving supply and demand.

Tips and Tricks

Here are some tips and tricks to keep in mind when solving systems of equations by substitution:

  • Make sure to substitute the correct equation into the other equation.
  • Simplify the equation by distributing and combining like terms.
  • Check your work by plugging the values back into the original equations.

Common Mistakes

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

  • Failing to substitute the correct equation into the other equation.
  • Not simplifying the equation by distributing and combining like terms.
  • Not checking your work by plugging the values back into the original equations.

Conclusion

Solving systems of equations by substitution is a powerful tool in mathematics. By following the steps and tips outlined in this article, you can solve systems of equations with ease. Remember to always check your work and avoid common mistakes.

Final Answer

The final answer is:

x=2x = 2 y=4y = 4

References

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

Related Topics

  • Solving Systems of Equations by Elimination
  • Solving Systems of Equations by Graphing
  • Systems of Linear Equations

Glossary

  • System of Equations: A set of two or more equations that are related to each other.
  • Substitution Method: A method of solving systems of equations by substituting one equation into another.
  • Linear Equation: An equation in which the highest power of the variable is 1.
  • Variable: A value that can change in a mathematical expression.
    Solving Systems of Equations by Substitution: Q&A =====================================================

Introduction

In our previous article, we discussed how to solve systems of equations by substitution. This method involves substituting one equation into another to solve for the variables. In this article, we will answer some frequently asked questions about solving systems of equations by substitution.

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of equations by substituting one equation into another. This method is useful when one of the equations is already solved for one of the variables.

Q: How do I know which equation to substitute into the other?

A: To determine which equation to substitute into the other, look for the equation that is already solved for one of the variables. This equation is usually the one that is easier to work with.

Q: What if I have two equations with two variables, and neither equation is solved for one of the variables?

A: In this case, you can use the elimination method to solve the system of equations. This method involves adding or subtracting the equations to eliminate one of the variables.

Q: Can I use the substitution method with non-linear equations?

A: No, the substitution method is typically used with linear equations. Non-linear equations require different techniques, such as the quadratic formula or graphing.

Q: How do I check my work when using the substitution method?

A: To check your work, plug the values back into the original equations to make sure they are true. This will help you verify that your solution is correct.

Q: What if I get a system of equations with no solution?

A: If you get a system of equations with no solution, it means that the equations are inconsistent. This can happen when the equations are contradictory, such as 2x + 3y = 5 and 2x + 3y = 10.

Q: Can I use the substitution method with systems of equations with more than two variables?

A: Yes, the substitution method can be used with systems of equations with more than two variables. However, it may be more complicated and require more steps.

Q: What are some common mistakes to avoid when using the substitution method?

A: Some common mistakes to avoid when using the substitution method include:

  • Failing to substitute the correct equation into the other equation.
  • Not simplifying the equation by distributing and combining like terms.
  • Not checking your work by plugging the values back into the original equations.

Q: How do I know when to use the substitution method versus the elimination method?

A: The choice between the substitution method and the elimination method depends on the specific system of equations. If one of the equations is already solved for one of the variables, the substitution method may be easier to use. If the equations are not easily solvable by substitution, the elimination method may be a better choice.

Conclusion

Solving systems of equations by substitution is a powerful tool in mathematics. By understanding the substitution method and how to use it, you can solve systems of equations with ease. Remember to always check your work and avoid common mistakes.

Final Answer

The final answer is:

  • The substitution method is a technique used to solve systems of equations by substituting one equation into another.
  • The substitution method is useful when one of the equations is already solved for one of the variables.
  • The substitution method can be used with linear equations, but not with non-linear equations.
  • The substitution method can be used with systems of equations with more than two variables.
  • The substitution method requires checking your work by plugging the values back into the original equations.

References

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

Related Topics

  • Solving Systems of Equations by Elimination
  • Solving Systems of Equations by Graphing
  • Systems of Linear Equations

Glossary

  • System of Equations: A set of two or more equations that are related to each other.
  • Substitution Method: A method of solving systems of equations by substituting one equation into another.
  • Linear Equation: An equation in which the highest power of the variable is 1.
  • Variable: A value that can change in a mathematical expression.