Solve The System Of Equations By Using Substitution.${ \begin{array}{l} y = 3x \ x + Y = -32 \end{array} }$(A) { (-8, -24)$}$ (B) { (-16, -48)$}$ (C) { (8, 24)$}$ (D) { (16, 48)$}$

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Introduction

Solving systems of equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will focus on solving systems of linear equations using the substitution method. This method involves solving one equation for a variable and then substituting that expression into the other equation to solve for the remaining variable.

What is the Substitution Method?

The substitution method is a technique used to solve systems of linear equations. It involves solving one equation for a variable and then substituting that expression into the other equation to solve for the remaining variable. This method is particularly useful when one of the equations is already solved for one of the variables.

Step-by-Step Guide to Solving Systems of Equations by Substitution

Step 1: Identify the Equations

The first step in solving a system of equations by substitution is to identify the two equations. In this case, we have two equations:

  1. y=3xy = 3x
  2. x+y=−32x + y = -32

Step 2: Solve One Equation for a Variable

The next step is to solve one of the equations for a variable. Let's solve the first equation for yy:

y=3xy = 3x

This equation is already solved for yy, so we can move on to the next step.

Step 3: Substitute the Expression into the Other Equation

Now that we have solved one equation for a variable, we can substitute that expression into the other equation. Let's substitute the expression for yy into the second equation:

x+(3x)=−32x + (3x) = -32

Step 4: Simplify the Equation

The next step is to simplify the equation by combining like terms:

4x=−324x = -32

Step 5: Solve for the Variable

Now that we have simplified the equation, we can solve for the variable xx:

x=−32/4x = -32/4

x=−8x = -8

Step 6: Find the Value of the Other Variable

Now that we have found the value of xx, we can substitute it into one of the original equations to find the value of the other variable. Let's substitute xx into the first equation:

y=3(−8)y = 3(-8)

y=−24y = -24

Step 7: Write the Solution as an Ordered Pair

The final step is to write the solution as an ordered pair. In this case, the solution is:

(−8,−24)(-8, -24)

Conclusion

Solving systems of equations by substitution is a powerful technique that can be used to solve a wide range of problems. By following the steps outlined in this article, you can solve systems of equations using the substitution method. Remember to identify the equations, solve one equation for a variable, substitute the expression into the other equation, simplify the equation, solve for the variable, find the value of the other variable, and write the solution as an ordered pair.

Answer

The correct answer is:

(A) (−8,−24)(-8, -24)

Discussion

Solving systems of equations is an important concept in mathematics, and it has numerous applications in various fields. The substitution method is a powerful technique that can be used to solve systems of equations. By following the steps outlined in this article, you can solve systems of equations using the substitution method.

Example Problems

Example Problem 1

Solve the system of equations using the substitution method:

y=2xx+y=10\begin{array}{l} y = 2x \\ x + y = 10 \end{array}

Solution

Step 1: Identify the equations

The two equations are:

  1. y=2xy = 2x
  2. x+y=10x + y = 10

Step 2: Solve one equation for a variable

The first equation is already solved for yy, so we can move on to the next step.

Step 3: Substitute the expression into the other equation

Let's substitute the expression for yy into the second equation:

x+(2x)=10x + (2x) = 10

Step 4: Simplify the equation

The next step is to simplify the equation by combining like terms:

3x=103x = 10

Step 5: Solve for the variable

Now that we have simplified the equation, we can solve for the variable xx:

x=10/3x = 10/3

Step 6: Find the value of the other variable

Now that we have found the value of xx, we can substitute it into one of the original equations to find the value of the other variable. Let's substitute xx into the first equation:

y=2(10/3)y = 2(10/3)

y=20/3y = 20/3

Step 7: Write the solution as an ordered pair

The final step is to write the solution as an ordered pair. In this case, the solution is:

(10/3,20/3)(10/3, 20/3)

Example Problem 2

Solve the system of equations using the substitution method:

y=x+2x+y=12\begin{array}{l} y = x + 2 \\ x + y = 12 \end{array}

Solution

Step 1: Identify the equations

The two equations are:

  1. y=x+2y = x + 2
  2. x+y=12x + y = 12

Step 2: Solve one equation for a variable

Let's solve the first equation for yy:

y=x+2y = x + 2

Step 3: Substitute the expression into the other equation

Now that we have solved one equation for a variable, we can substitute that expression into the other equation. Let's substitute the expression for yy into the second equation:

x+(x+2)=12x + (x + 2) = 12

Step 4: Simplify the equation

The next step is to simplify the equation by combining like terms:

2x+2=122x + 2 = 12

Step 5: Solve for the variable

Now that we have simplified the equation, we can solve for the variable xx:

2x=102x = 10

x=5x = 5

Step 6: Find the value of the other variable

Now that we have found the value of xx, we can substitute it into one of the original equations to find the value of the other variable. Let's substitute xx into the first equation:

y=5+2y = 5 + 2

y=7y = 7

Step 7: Write the solution as an ordered pair

The final step is to write the solution as an ordered pair. In this case, the solution is:

(5,7)(5, 7)

Practice Problems

Practice Problem 1

Solve the system of equations using the substitution method:

y=4xx+y=20\begin{array}{l} y = 4x \\ x + y = 20 \end{array}

Practice Problem 2

Solve the system of equations using the substitution method:

y=2x−3x+y=7\begin{array}{l} y = 2x - 3 \\ x + y = 7 \end{array}

Practice Problem 3

Solve the system of equations using the substitution method:

y=x−2x+y=9\begin{array}{l} y = x - 2 \\ x + y = 9 \end{array}

Practice Problem 4

Solve the system of equations using the substitution method:

y=3x+1x+y=14\begin{array}{l} y = 3x + 1 \\ x + y = 14 \end{array}

Answer Key

Answer Key for Practice Problems

Practice Problem 1

The solution is:

(5,20)(5, 20)

Practice Problem 2

The solution is:

(5,7)(5, 7)

Practice Problem 3

The solution is:

(11,−2)(11, -2)

Practice Problem 4

The solution is:

Introduction

Solving systems of equations by substitution is a powerful technique that can be used to solve a wide range of problems. In this article, we will answer some of the most 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 linear equations. It involves solving one equation for a variable and then substituting that expression into the other equation to solve for the remaining variable.

Q: How do I know which equation to solve for first?

A: You can choose either equation to solve for first. However, it is often easier to solve for the variable that appears in both equations.

Q: What if I get stuck during the substitution process?

A: If you get stuck during the substitution process, try to simplify the equation by combining like terms. If you are still having trouble, try to visualize the problem and see if you can come up with a different approach.

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

A: No, the substitution method is only used to solve systems of linear equations. If you have a system of nonlinear equations, you will need to use a different method, such as the elimination method or the graphing method.

Q: How do I know if I have found the correct solution?

A: To check if you have found the correct solution, plug the values of the variables back into both original equations. If the equations are true, then you have found the correct solution.

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 if the equations are contradictory, such as 2x + 3 = 5 and 2x + 3 = 7.

Q: Can I use the substitution method to solve systems of equations with three variables?

A: Yes, you can use the substitution method to solve systems of equations with three variables. However, it may be more difficult to solve the system, and you may need to use a different method, such as the elimination method or the graphing method.

Q: How do I choose the correct method to solve a system of equations?

A: The choice of method depends on the type of equations and the number of variables. If you have a system of linear equations with two variables, the substitution method is often the easiest and most efficient method to use. However, if you have a system of nonlinear equations or a system with three variables, you may need to use a different method.

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

A: Yes, you can use the substitution method to solve systems of equations with fractions or decimals. However, you will need to be careful when simplifying the equations and solving for the variables.

Q: How do I know if I have found the correct solution when working with fractions or decimals?

A: To check if you have found the correct solution when working with fractions or decimals, plug the values of the variables back into both original equations. If the equations are true, then you have found the correct solution.

Conclusion

Solving systems of equations by substitution is a powerful technique that can be used to solve a wide range of problems. By following the steps outlined in this article and answering the frequently asked questions, you can become proficient in solving systems of equations by substitution.

Practice Problems

Practice Problem 1

Solve the system of equations using the substitution method:

y=2xx+y=10\begin{array}{l} y = 2x \\ x + y = 10 \end{array}

Practice Problem 2

Solve the system of equations using the substitution method:

y=x+2x+y=12\begin{array}{l} y = x + 2 \\ x + y = 12 \end{array}

Practice Problem 3

Solve the system of equations using the substitution method:

y=3x−2x+y=11\begin{array}{l} y = 3x - 2 \\ x + y = 11 \end{array}

Practice Problem 4

Solve the system of equations using the substitution method:

y=2x+1x+y=13\begin{array}{l} y = 2x + 1 \\ x + y = 13 \end{array}

Answer Key

Answer Key for Practice Problems

Practice Problem 1

The solution is:

(5,10)(5, 10)

Practice Problem 2

The solution is:

(5,7)(5, 7)

Practice Problem 3

The solution is:

(7,4)(7, 4)

Practice Problem 4

The solution is:

(6,11)(6, 11)