Solving a System of Equations for Two Numbers

In this article, we will solve a system of equations to find two numbers that satisfy the given conditions. This type of problem is common in algebra and can be solved using various methods. We will explore a few different approaches to solve the system of equations and verify the solutions.

Problem Statement

The sum of two numbers is 58. If twice the smaller number is subtracted from the larger number, the result is 7. What are the two numbers?

Solution Methods

Method 1: Substitution

Let the smaller number be x and the larger number be 58 - x.

Step 1: Substituting the larger number into the equation:

58 - x - 2x 7

Step 2: Simplify the equation:

58 - 3x 7

Step 3: Isolate x by moving all terms involving x to one side and constants to the other side:

3x 58 - 7

Step 4: Simplify further:

3x 51

Step 5: Divide both sides by 3 to solve for x:

x 17

Step 6: Find the larger number:

58 - 17 41

Conclusion: The two numbers are 17 and 41.

Method 2: Substitution and Simplification

Let the smaller number x and the larger number 58 - x.

58 - x - 2x 7

Step 1: Combine like terms:

58 - 3x 7

Step 2: Isolate x by simplifying:

3x 58 - 7

Step 3: Simplify further:

3x 51

Step 4: Divide both sides by 3 to solve for x:

x 17

Step 5: Find the larger number:

58 - 17 41

Conclusion: The two numbers are 17 and 41.

Method 3: Algebraic Substitution

Let x and y be the two numbers.

x y 58 and y - 2x 7

Step 1: Express y in terms of x from the first equation:

y 58 - x

Step 2: Substitute y into the second equation:

(58 - x) - 2x 7

Step 3: Simplify:

58 - 3x 7

Step 4: Isolate x:

3x 51

Step 5: Solve for x:

x 17

Step 6: Find y using the first equation:

17 y 58

y 41

Conclusion: The two numbers are 17 and 41.

Conclusion

By solving the system of equations, we have determined that the two numbers are 17 and 41. This problem demonstrates the application of algebraic methods to solve real-world scenarios involving a system of equations. Understanding these methods is crucial for solving more complex mathematical problems and is a fundamental skill in algebra.