Solve By The Substitution Method:$\[ \begin{aligned} 8x + 3y & = -1 \\ -8x + Y & = 21 \end{aligned} \\]What Is The Solution Of The System? (Type An Ordered Pair.)

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Introduction

In mathematics, systems of linear equations are a set of two or more linear equations that involve the same set of variables. Solving systems of linear equations is an essential skill in algebra and is used in various fields such as physics, engineering, and economics. There are several methods to solve systems of linear equations, including the substitution method, the elimination method, and the graphing method. In this article, we will focus on the substitution method and provide a step-by-step solution to the system of linear equations given below.

The Substitution Method

The substitution method is a technique used to solve systems of linear equations by substituting one equation into the other. This method involves solving one equation for one variable and then substituting that expression into the other equation. The resulting equation is then solved for the remaining variable.

Step 1: Solve One Equation for One Variable

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

−8x+y=21-8x + y = 21

We can isolate y by adding 8x to both sides of the equation:

y=8x+21y = 8x + 21

Step 2: Substitute the Expression into the Other Equation

Now that we have an expression for y, we can substitute it into the first equation:

8x+3y=−18x + 3y = -1

Substituting y = 8x + 21 into the first equation, we get:

8x+3(8x+21)=−18x + 3(8x + 21) = -1

Step 3: Simplify the Equation

Expanding the equation, we get:

8x+24x+63=−18x + 24x + 63 = -1

Combine like terms:

32x+63=−132x + 63 = -1

Subtract 63 from both sides:

32x=−6432x = -64

Divide both sides by 32:

x=−2x = -2

Step 4: Find the Value of y

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the second equation:

−8x+y=21-8x + y = 21

Substituting x = -2, we get:

−8(−2)+y=21-8(-2) + y = 21

Simplifying the equation, we get:

16+y=2116 + y = 21

Subtract 16 from both sides:

y=5y = 5

Conclusion

The solution to the system of linear equations is x = -2 and y = 5. Therefore, the ordered pair that satisfies both equations is (-2, 5).

Example Use Cases

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

  • Physics: When solving problems involving motion, such as the trajectory of a projectile, you may need to solve a system of linear equations to find the position and velocity of the object.
  • Engineering: In engineering, systems of linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: In economics, systems of linear equations are used to model economic systems and make predictions about the behavior of markets.

Tips and Tricks

Here are a few tips and tricks to help you solve systems of linear equations using the substitution method:

  • Choose the correct equation: When choosing which equation to solve for one variable, choose the one that is easiest to solve.
  • Check your work: Always check your work by plugging the values back into both equations to make sure they are true.
  • Use a calculator: If you are having trouble solving the equations by hand, try using a calculator to check your work.

Conclusion

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of linear equations by substituting one equation into the other. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Q: When should I use the substitution method?

A: You should use the substitution method when one of the equations is easier to solve for one variable than the other equation. This method is particularly useful when one of the equations is already solved for one variable.

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

A: When choosing which equation to solve for one variable, choose the one that is easiest to solve. If one equation has a coefficient of 1 for one variable, it is usually easier to solve for that variable.

Q: What if I get stuck while solving the system?

A: If you get stuck while solving the system, try the following:

  • Check your work: Make sure you have not made any mistakes in your calculations.
  • Use a calculator: If you are having trouble solving the equations by hand, try using a calculator to check your work.
  • Ask for help: If you are still having trouble, ask a teacher or tutor for help.

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

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

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 checking your work: Make sure you have not made any mistakes in your calculations.
  • Not using the correct equation: Choose the correct equation to solve for one variable.
  • Not simplifying the equation: Make sure to simplify the equation after substituting the expression.

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

A: To check if the solution is correct, plug the values back into both equations to make sure they are true. If the values satisfy both equations, then the solution is correct.

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

A: Yes, you can use the substitution method to solve systems of linear equations with fractions or decimals. However, you may need to use additional techniques, such as multiplying both sides of the equation by a common denominator, to eliminate the fractions or decimals.

Q: What are some real-world applications of the substitution method?

A: The substitution method has many real-world applications, including:

  • Physics: When solving problems involving motion, such as the trajectory of a projectile, you may need to solve a system of linear equations to find the position and velocity of the object.
  • Engineering: In engineering, systems of linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: In economics, systems of linear equations are used to model economic systems and make predictions about the behavior of markets.

Conclusion

In conclusion, the substitution method is a powerful tool for solving systems of linear equations. By following the steps outlined in this article and avoiding common mistakes, you can solve systems of linear equations and find the solution to the system. Remember to choose the correct equation, check your work, and use a calculator if needed. With practice and patience, you will become proficient in solving systems of linear equations using the substitution method.