To Solve This System Of Equations Using Substitution, What Could Be Substituted In Place Of $y$ In The First Equation?${ \begin{array}{l} 4x = 5 - 2y \ y - 2x = 7 \end{array} }$A. $7 - 2x$ B. $4x - 5$ C.

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Introduction

Solving systems of equations is a fundamental concept in mathematics, and one of the most effective methods is substitution. In this article, we will explore how to use substitution to solve a system of equations, focusing on the first equation and identifying what can be substituted in place of y.

Understanding the System of Equations

The given system of equations is:

{ \begin{array}{l} 4x = 5 - 2y \\ y - 2x = 7 \end{array} \}

To solve this system using substitution, we need to isolate one of the variables in one of the equations. In this case, we can start by isolating y in the second equation.

Isolating y in the Second Equation

The second equation is:

y - 2x = 7

To isolate y, we can add 2x to both sides of the equation:

y = 7 + 2x

Now, we have y expressed in terms of x.

Substituting y in the First Equation

The first equation is:

4x = 5 - 2y

We can substitute the expression for y from the second equation into the first equation:

4x = 5 - 2(7 + 2x)

To simplify this expression, we can distribute the -2 to both terms inside the parentheses:

4x = 5 - 14 - 4x

Now, we can combine like terms:

4x = -9 - 4x

Next, we can add 4x to both sides of the equation to get:

8x = -9

Finally, we can divide both sides of the equation by 8 to solve for x:

x = -9/8

What Can Be Substituted in Place of y?

Now that we have solved for x, we can substitute this value back into the expression for y from the second equation:

y = 7 + 2x

Substituting x = -9/8 into this expression, we get:

y = 7 + 2(-9/8)

To simplify this expression, we can multiply 2 by -9/8:

y = 7 - 9/4

Now, we can convert 7 to a fraction with a denominator of 4:

y = 28/4 - 9/4

Finally, we can combine like terms:

y = 19/4

Conclusion

In this article, we have explored how to use substitution to solve a system of equations. We started by isolating y in the second equation and then substituted this expression into the first equation. By following these steps, we were able to solve for x and then substitute this value back into the expression for y. The final answer is y = 19/4.

Discussion

  • What are some other methods for solving systems of equations?
  • How can substitution be used to solve systems of equations with more than two variables?
  • What are some common pitfalls to avoid when using substitution to solve systems of equations?

References

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

Additional Resources

  • [1] "Systems of Equations" by MIT OpenCourseWare
  • [2] "Substitution Method" by Purplemath
    Frequently Asked Questions: Solving Systems of Equations Using Substitution ====================================================================

Q: What is substitution in solving systems of equations?

A: Substitution is a method of solving systems of equations by expressing one variable in terms of another variable and then substituting that expression into the other equation.

Q: How do I choose which variable to substitute?

A: You can choose either variable to substitute, but it's often easier to substitute the variable that appears in both equations. In the example we used earlier, we chose to substitute y in the first equation.

Q: What if I have a system of equations with more than two variables?

A: In that case, you can use substitution to solve for one variable in terms of the other variables, and then substitute that expression into the other equations. You can repeat this process until you have solved for all the variables.

Q: What are some common pitfalls to avoid when using substitution?

A: Some common pitfalls to avoid when using substitution include:

  • Not isolating one of the variables in one of the equations before substituting.
  • Not distributing the coefficient to both terms inside the parentheses when substituting.
  • Not combining like terms correctly when simplifying the equation.

Q: Can I use substitution to solve systems of equations with fractions?

A: Yes, you can use substitution to solve systems of equations with fractions. Just be sure to simplify the fractions correctly when substituting and combining like terms.

Q: How do I know if I have solved the system of equations correctly?

A: To check if you have solved the system of equations correctly, you can substitute the values of the variables back into both original equations and make sure they are true.

Q: What are some other methods for solving systems of equations?

A: Some other methods for solving systems of equations include:

  • The addition method: This method involves adding the two equations together to eliminate one of the variables.
  • The multiplication method: This method involves multiplying one or both of the equations by a constant to eliminate one of the variables.
  • Graphing: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.

Q: Can I use substitution to solve systems of equations with nonlinear equations?

A: Yes, you can use substitution to solve systems of equations with nonlinear equations. However, you may need to use more advanced techniques, such as implicit differentiation, to solve for the variables.

Q: How do I choose which method to use to solve a system of equations?

A: The choice of method depends on the specific system of equations and the variables involved. You may need to try different methods to see which one works best.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of equations using substitution. We have covered topics such as choosing which variable to substitute, common pitfalls to avoid, and other methods for solving systems of equations. By following these tips and techniques, you can become proficient in solving systems of equations using substitution.

Discussion

  • What are some other methods for solving systems of equations that you have used?
  • How do you choose which method to use to solve a system of equations?
  • What are some common pitfalls to avoid when using substitution?

References

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

Additional Resources

  • [1] "Systems of Equations" by MIT OpenCourseWare
  • [2] "Substitution Method" by Purplemath