The Product Of Two Numbers Is 54, And Their Difference Is 3. Find The Numbers.

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Introduction

In this problem, we are given two pieces of information about two numbers: their product and their difference. We will use algebraic methods to find the two numbers that satisfy these conditions.

Problem Statement

Let's denote the two numbers as x and y. We are given that their product is 54, which can be written as:

xy = 54

We are also given that their difference is 3, which can be written as:

x - y = 3

Algebraic Methods

To solve this problem, we can use algebraic methods to find the values of x and y. One way to do this is to use the method of substitution.

Method of Substitution

We can start by solving the second equation for x:

x = y + 3

Now, we can substitute this expression for x into the first equation:

(y + 3)y = 54

Expanding the left-hand side of the equation, we get:

y^2 + 3y = 54

Rearranging the equation to form a quadratic equation, we get:

y^2 + 3y - 54 = 0

Solving the Quadratic Equation

To solve this quadratic equation, we can use the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = 3, and c = -54. Plugging these values into the formula, we get:

y = (-3 ± √(3^2 - 4(1)(-54))) / 2(1)

Simplifying the expression under the square root, we get:

y = (-3 ± √(9 + 216)) / 2

y = (-3 ± √225) / 2

y = (-3 ± 15) / 2

Finding the Values of y

Now, we can find the two possible values of y by solving the two equations:

y = (-3 + 15) / 2

y = 12 / 2

y = 6

y = (-3 - 15) / 2

y = -18 / 2

y = -9

Finding the Values of x

Now that we have found the values of y, we can find the corresponding values of x by using the expression:

x = y + 3

For y = 6, we get:

x = 6 + 3

x = 9

For y = -9, we get:

x = -9 + 3

x = -6

Conclusion

In this problem, we used algebraic methods to find the two numbers that satisfy the given conditions. We found that the two numbers are 9 and 6, or -6 and -9.

Final Answer

The final answer is:

  • x = 9, y = 6
  • x = -6, y = -9

Discussion

This problem is a classic example of a system of linear equations. We used algebraic methods to solve the system and find the values of x and y. The method of substitution is a powerful tool for solving systems of linear equations.

Related Problems

This problem is related to other problems in algebra, such as solving systems of linear equations and quadratic equations. It is also related to other areas of mathematics, such as geometry and trigonometry.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Linear Algebra and Its Applications" by Gilbert Strang
  • [3] "Calculus" by Michael Spivak

Keywords

  • Algebra
  • Linear equations
  • Quadratic equations
  • System of linear equations
  • Method of substitution
  • Algebraic methods
  • Mathematics
  • Geometry
  • Trigonometry

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Introduction

In our previous article, we solved the problem of finding two numbers whose product is 54 and whose difference is 3. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the product of two numbers?

A: The product of two numbers is the result of multiplying them together. In this problem, the product of the two numbers is 54.

Q: What is the difference of two numbers?

A: The difference of two numbers is the result of subtracting one number from the other. In this problem, the difference of the two numbers is 3.

Q: How do we find the two numbers?

A: We can use algebraic methods to find the two numbers. One way to do this is to use the method of substitution.

Q: What is the method of substitution?

A: The method of substitution is a technique used to solve systems of linear equations. We substitute one equation into another equation to eliminate one of the variables.

Q: How do we use the method of substitution to solve this problem?

A: We start by solving one of the equations for one of the variables. Then, we substitute that expression into the other equation. We can then solve for the other variable.

Q: What are the two possible values of y?

A: The two possible values of y are 6 and -9.

Q: What are the corresponding values of x?

A: The corresponding values of x are 9 and -6.

Q: What is the final answer?

A: The final answer is:

  • x = 9, y = 6
  • x = -6, y = -9

Common Mistakes

Mistake 1: Not using the correct method

A: One common mistake is not using the correct method to solve the problem. In this case, we should use the method of substitution.

Mistake 2: Not checking the solutions

A: Another common mistake is not checking the solutions to make sure they satisfy both equations.

Mistake 3: Not using the correct algebraic techniques

A: A third common mistake is not using the correct algebraic techniques to solve the problem.

Tips and Tricks

Tip 1: Use the correct method

A: Use the correct method to solve the problem. In this case, we should use the method of substitution.

Tip 2: Check the solutions

A: Check the solutions to make sure they satisfy both equations.

Tip 3: Use the correct algebraic techniques

A: Use the correct algebraic techniques to solve the problem.

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding two numbers whose product is 54 and whose difference is 3. We also discussed some common mistakes and provided some tips and tricks for solving the problem.

Final Answer

The final answer is:

  • x = 9, y = 6
  • x = -6, y = -9

Discussion

This problem is a classic example of a system of linear equations. We used algebraic methods to solve the system and find the values of x and y. The method of substitution is a powerful tool for solving systems of linear equations.

Related Problems

This problem is related to other problems in algebra, such as solving systems of linear equations and quadratic equations. It is also related to other areas of mathematics, such as geometry and trigonometry.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Linear Algebra and Its Applications" by Gilbert Strang
  • [3] "Calculus" by Michael Spivak

Keywords

  • Algebra
  • Linear equations
  • Quadratic equations
  • System of linear equations
  • Method of substitution
  • Algebraic methods
  • Mathematics
  • Geometry
  • Trigonometry