Sam (S) Is 9 Years Older Than Victor (V). The Sum Of Their Ages Is 33. Find The Age Of Each Person.${ S = [?] \quad V = [?] }$

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Introduction

In this article, we will explore a simple yet effective problem in mathematics that involves solving a system of linear equations. The problem states that Sam (S) is 9 years older than Victor (V), and the sum of their ages is 33. We will use algebraic methods to find the age of each person.

Problem Statement

Let's denote Sam's age as S and Victor's age as V. We are given two pieces of information:

  1. Sam is 9 years older than Victor: S = V + 9
  2. The sum of their ages is 33: S + V = 33

Setting Up the System of Linear Equations

We can represent the two equations as a system of linear equations:

S = V + 9 ... (Equation 1) S + V = 33 ... (Equation 2)

Solving the System of Linear Equations

To solve this system, we can use the method of substitution or elimination. Let's use the substitution method.

Step 1: Substitute Equation 1 into Equation 2

We can substitute the expression for S from Equation 1 into Equation 2:

(V + 9) + V = 33

Step 2: Simplify the Equation

Combine like terms:

2V + 9 = 33

Step 3: Isolate the Variable

Subtract 9 from both sides:

2V = 24

Step 4: Solve for the Variable

Divide both sides by 2:

V = 12

Step 5: Find the Value of S

Now that we have found V, we can substitute this value into Equation 1 to find S:

S = V + 9 S = 12 + 9 S = 21

Conclusion

We have successfully solved the system of linear equations to find the age of each person. Sam is 21 years old, and Victor is 12 years old.

Final Answer

S=21V=12{ S = 21 \quad V = 12 }

Discussion

This problem is a great example of how algebraic methods can be used to solve real-world problems. The system of linear equations can be represented graphically as two lines intersecting at a single point, which represents the solution to the system.

Tips and Variations

  • To make this problem more challenging, you can add more variables or equations to the system.
  • You can also use different methods, such as the elimination method, to solve the system.
  • This problem can be extended to find the ages of multiple people, given their relationships and total age.

Related Topics

  • Systems of linear equations
  • Algebraic methods
  • Problem-solving strategies

References

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Introduction

In our previous article, we explored a simple yet effective problem in mathematics that involved solving a system of linear equations. The problem stated that Sam (S) is 9 years older than Victor (V), and the sum of their ages is 33. We used algebraic methods to find the age of each person. In this article, we will answer some frequently asked questions related to solving systems of linear equations.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: How do I solve a system of linear equations?

There are several methods to solve a system of linear equations, including the substitution method, elimination method, and graphing method. The substitution method involves substituting one equation into another, while the elimination method involves adding or subtracting equations to eliminate one of the variables.

Q: What is the substitution method?

The substitution method involves substituting one equation into another to solve for one of the variables. This method is useful when one of the equations is already solved for one of the variables.

Q: What is the elimination method?

The elimination method involves adding or subtracting equations to eliminate one of the variables. This method is useful when the coefficients of one of the variables are the same in both equations.

Q: How do I choose the correct method?

The choice of method depends on the type of equations and the variables involved. If one of the equations is already solved for one of the variables, the substitution method may be the best choice. If the coefficients of one of the variables are the same in both equations, the elimination method may be the best choice.

Q: What are some common mistakes to avoid?

Some common mistakes to avoid when solving systems of linear equations include:

  • Not checking the solution to ensure it satisfies both equations
  • Not using the correct method for the type of equations
  • Not simplifying the equations before solving
  • Not checking for extraneous solutions

Q: How do I check the solution?

To check the solution, substitute the values of the variables back into both equations to ensure that they are true.

Q: What are some real-world applications of systems of linear equations?

Systems of linear equations have many real-world applications, including:

  • Finance: solving systems of linear equations can help with budgeting and financial planning
  • Science: solving systems of linear equations can help with modeling and predicting scientific phenomena
  • Engineering: solving systems of linear equations can help with designing and optimizing systems

Q: How do I practice solving systems of linear equations?

There are many resources available to practice solving systems of linear equations, including:

  • Online tutorials and videos
  • Practice problems and worksheets
  • Math textbooks and workbooks

Q: What are some advanced topics in systems of linear equations?

Some advanced topics in systems of linear equations include:

  • Systems of nonlinear equations
  • Systems of differential equations
  • Matrix algebra

Q: How do I learn more about systems of linear equations?

There are many resources available to learn more about systems of linear equations, including:

  • Online courses and tutorials
  • Math textbooks and workbooks
  • Math communities and forums

Conclusion

Solving systems of linear equations is a fundamental skill in mathematics that has many real-world applications. By understanding the different methods and techniques, you can solve a wide range of problems and make informed decisions. Remember to practice regularly and seek help when needed to become proficient in solving systems of linear equations.

Final Answer

S=21V=12{ S = 21 \quad V = 12 }

Discussion

This article provides a comprehensive overview of solving systems of linear equations, including the substitution method, elimination method, and graphing method. It also covers common mistakes to avoid, real-world applications, and advanced topics.

Tips and Variations

  • To make this problem more challenging, you can add more variables or equations to the system.
  • You can also use different methods, such as the elimination method, to solve the system.
  • This problem can be extended to find the ages of multiple people, given their relationships and total age.

Related Topics

  • Systems of linear equations
  • Algebraic methods
  • Problem-solving strategies

References