Solve The System Of Equations By Substitution.${ \begin{align*} y &= \frac{1}{4}x - 1 \ 2y &= \frac{2}{3}x + 6 \end{align*} }$

by ADMIN 128 views

Introduction

Solving a system of equations is a fundamental concept in mathematics, and it involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of linear equations using the substitution method. The system of equations we will be working with is:

{ \begin{align*} y &= \frac{1}{4}x - 1 \\ 2y &= \frac{2}{3}x + 6 \end{align*} \}

Understanding the Substitution Method

The substitution method is a technique used to solve a system of equations by substituting one equation into the other. This method is particularly useful when one of the equations is already solved for one of the variables. In our case, the first equation is already solved for y, which makes it an ideal candidate for substitution.

Step 1: Solve One of the Equations for a Variable

Let's start by solving the first equation for y:

{ y = \frac{1}{4}x - 1 \}

This equation is already solved for y, so we can proceed to the next step.

Step 2: Substitute the Expression into the Other Equation

Now, let's substitute the expression for y into the second equation:

{ 2y = \frac{2}{3}x + 6 \}

Substituting the expression for y, we get:

{ 2(\frac{1}{4}x - 1) = \frac{2}{3}x + 6 \}

Step 3: Simplify the Equation

Now, let's simplify the equation by distributing the 2:

{ \frac{1}{2}x - 2 = \frac{2}{3}x + 6 \}

Step 4: Isolate the Variable

Next, let's isolate the variable x by moving all the terms involving x to one side of the equation:

{ \frac{1}{2}x - \frac{2}{3}x = 6 + 2 \}

Combining like terms, we get:

{ -\frac{1}{6}x = 8 \}

Step 5: Solve for the Variable

Finally, let's solve for x by multiplying both sides of the equation by -6:

{ x = -48 \}

Step 6: Find the Value of the Other Variable

Now that we have the value of x, let's substitute it back into one of the original equations to find the value of y. We'll use the first equation:

{ y = \frac{1}{4}x - 1 \}

Substituting x = -48, we get:

{ y = \frac{1}{4}(-48) - 1 \}

Simplifying, we get:

{ y = -12 - 1 \}

{ y = -13 \}

Conclusion

In this article, we solved a system of linear equations using the substitution method. We started by solving one of the equations for a variable, then substituted the expression into the other equation. We simplified the equation, isolated the variable, and finally solved for the variable. We also found the value of the other variable by substituting the value of the first variable back into one of the original equations. The final answer is x = -48 and y = -13.

Example Use Cases

The substitution method is a powerful tool for solving systems of equations. Here are a few example use cases:

  • Physics: When solving problems involving motion, you may need to use the substitution method to find the values of variables such as velocity and acceleration.
  • Engineering: In engineering, you may need to use the substitution method to solve systems of equations that describe the behavior of complex systems.
  • Computer Science: In computer science, you may need to use the substitution method to solve systems of equations that arise in the context of algorithms and data structures.

Tips and Tricks

Here are a few tips and tricks to keep in mind when using the substitution method:

  • Choose the right equation: When choosing which equation to solve for a variable, choose the one that is easiest to solve.
  • Substitute carefully: When substituting an expression into an equation, make sure to substitute it correctly.
  • Simplify the equation: When simplifying the equation, make sure to combine like terms and eliminate any unnecessary variables.

Conclusion

Q: What is the substitution method?

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

Q: When should I use the substitution method?

A: You should use the substitution method when one of the equations is already solved for one of the variables. This makes it easier to substitute the expression into the other equation.

Q: How do I choose which equation to solve for a variable?

A: When choosing which equation to solve for a variable, choose the one that is easiest to solve. This will make the substitution process easier and less prone to errors.

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 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: What if I make a mistake during the substitution process?

A: If you make a mistake during the substitution process, go back and recheck your work. Make sure to substitute the expression correctly and simplify the equation carefully.

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

A: Yes, you can use the substitution method with systems of equations with more than two variables. However, this can become increasingly complex and may require additional techniques, such as the elimination method.

Q: How do I know if the solution is correct?

A: To verify the solution, plug the values back into the original equations and check if they are true. If the values satisfy both equations, then the solution is correct.

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:

  • Not substituting the expression correctly
  • Not simplifying the equation carefully
  • Not checking the solution against the original equations
  • Not using the correct method for the type of equation (e.g., using the substitution method for non-linear equations)

Q: Can I use the substitution method with systems of equations with fractions or decimals?

A: Yes, you can use the substitution method with systems of equations with fractions or decimals. However, be careful when simplifying the equation and make sure to handle the fractions or decimals correctly.

Q: How do I apply the substitution method to real-world problems?

A: To apply the substitution method to real-world problems, identify the variables and equations involved, and then use the substitution method to solve the system of equations. Make sure to check the solution against the original problem to ensure that it is correct.

Conclusion

In conclusion, the substitution method is a powerful tool for solving systems of equations. By following the steps outlined in this article and avoiding common mistakes, you can use the substitution method to solve systems of equations with confidence. Remember to choose the right equation, substitute carefully, and simplify the equation to get the correct solution. With practice and patience, you can become proficient in using the substitution method to solve systems of equations.