Expressing the Solution Set in Terms of One Variable

Oct 13, 2025 by miodeya.com 101 views

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. The substitution method is one of the techniques used to solve a system of equations. This method involves substituting the expression for one variable from one equation into the other equation. In this article, we will use the substitution method to solve a system of equations and express the solution set in terms of one of the variables.

The System of Equations

The given system of equations is:

$\[ \begin{cases} 5x + y = 40 \\ x = 30 \end{cases}$

Step 1: Identify the Equation to Substitute

In this system of equations, we have two equations. The second equation is already solved for x, which is x = 30. We can substitute this value of x into the first equation to solve for y.

Step 2: Substitute the Value of x into the First Equation

Substitute x = 30 into the first equation:

5x + y = 40

Replace x with 30:

5(30) + y = 40

Step 3: Simplify the Equation

Simplify the equation:

150 + y = 40

Step 4: Solve for y

Subtract 150 from both sides of the equation:

y = -110

Conclusion

The solution to the system of equations is x = 30 and y = -110. Since the system has a unique solution, it is not dependent.

Expressing the Solution Set in Terms of One Variable

In this case, we can express the solution set in terms of x. Since x = 30, we can substitute this value of x into the first equation to get:

5(30) + y = 40

Simplify the equation:

150 + y = 40

Subtract 150 from both sides of the equation:

y = -110

Therefore, the solution set can be expressed as:

x = 30, y = -110

Discussion

The substitution method is a powerful technique for solving systems of equations. By substituting the expression for one variable from one equation into the other equation, we can solve for the values of the variables. In this case, we used the substitution method to solve a system of equations and express the solution set in terms of one of the variables.

Example

Let's consider another example of a system of equations:

$\[ \begin{cases} 2x + 3y = 12 \\ x = 4 \end{cases}$

We can use the substitution method to solve this system of equations. Substitute x = 4 into the first equation:

2x + 3y = 12

Replace x with 4:

2(4) + 3y = 12

Simplify the equation:

8 + 3y = 12

Subtract 8 from both sides of the equation:

3y = 4

Divide both sides of the equation by 3:

y = 4/3

Therefore, the solution to the system of equations is x = 4 and y = 4/3.

Conclusion

In conclusion, the substitution method is a useful technique for solving systems of equations. By substituting the expression for one variable from one equation into the other equation, we can solve for the values of the variables. In this article, we used the substitution method to solve a system of equations and express the solution set in terms of one of the variables.

References

[1] "Systems of Equations" by Math Open Reference. Retrieved from https://www.mathopenref.com/systems.html
[2] "Substitution Method" by Purplemath. Retrieved from https://www.purplemath.com/modules/substns.htm

Glossary

System of Equations: A set of two or more equations that are solved simultaneously to find the values of the variables.
Substitution Method: A technique used to solve a system of equations by substituting the expression for one variable from one equation into the other equation.
Dependent System: A system of equations that has an infinite number of solutions.
Independent System: A system of equations that has a unique solution.

Introduction

In our previous article, we discussed how to solve a system of equations using the substitution method. In this article, we will answer some frequently asked questions about solving systems of equations using the substitution method.

Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of equations by substituting the expression for one variable from one equation into the other equation.

Q: When can I use the substitution method?

A: You can use the substitution method when one of the equations is already solved for one of the variables. This is often the case when one of the equations is an identity or a simple equation.

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

A: You should substitute the equation that is already solved for one of the variables into the other equation. This will allow you to solve for the other variable.

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

A: In this case, you can use the elimination method or the substitution method after first solving one of the equations for one of the variables.

Q: Can I use the substitution method to solve a system of equations with more than two variables?

A: Yes, you can use the substitution method to solve a system of equations with more than two variables. However, it may be more complicated and require more steps.

Q: What if I get a fraction or a decimal as a solution?

A: If you get a fraction or a decimal as a solution, you should leave it in that form. Do not simplify or approximate the solution unless instructed to do so.

Q: Can I use the substitution method to solve a system of equations with dependent variables?

A: No, the substitution method is not suitable for solving a system of equations with dependent variables. In this case, you should use the elimination method or the graphical method.

Q: How do I know if a system of equations has a dependent or independent solution?

A: If the system of equations has a unique solution, it is an independent system. If the system of equations has an infinite number of solutions, it is a dependent system.

Q: Can I use the substitution method to solve a system of equations with complex numbers?

A: Yes, you can use the substitution method to solve a system of equations with complex numbers. However, you should be careful when working with complex numbers and follow the rules of complex arithmetic.

Q: What if I make a mistake while using the substitution method?

A: If you make a mistake while using the substitution method, you should go back and recheck your work. Make sure to follow the steps carefully and check your solutions.

Conclusion

In conclusion, the substitution method is a useful technique for solving systems of equations. By substituting the expression for one variable from one equation into the other equation, we can solve for the values of the variables. We hope that this article has answered some of the frequently asked questions about solving systems of equations using the substitution method.

References

[1] "Systems of Equations" by Math Open Reference. Retrieved from https://www.mathopenref.com/systems.html
[2] "Substitution Method" by Purplemath. Retrieved from https://www.purplemath.com/modules/substns.htm

Glossary

System of Equations: A set of two or more equations that are solved simultaneously to find the values of the variables.
Substitution Method: A technique used to solve a system of equations by substituting the expression for one variable from one equation into the other equation.
Dependent System: A system of equations that has an infinite number of solutions.
Independent System: A system of equations that has a unique solution.
Complex Number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.