Solving Equations: A Real-World Example of a System of Linear Equations

Equations are an essential part of algebra, often used to solve real-world problems involving multiple relationships. A common scenario involves expressing one variable in terms of another and then solving for the variables by combining these expressions. Let's explore a practical example that involves the relationship between two numbers and how to solve it using a system of linear equations.

Problem Statement and Definitions

The problem at hand states that one number is 7 more than twice another number. Simultaneously, the sum of these two numbers is 22. Mathematically, we can denote the two numbers as (x) and (y), where:

(x 2y 7) (x y 22)

Solving the Equations Step-by-Step

Substitution Method

Substituting the expression for (x) from the first equation into the second equation, we start with the given equations:

(x 2y 7) (x y 22)

Substitute (x 2y 7) into the second equation:

(2y 7 y 22)

Combine like terms:

(3y 7 22)

Subtract 7 from both sides:

(3y 15)

Divide by 3:

(y 5)

Now, substitute (y 5) back into the first equation to find (x):

(x 2(5) 7 10 7 17)

Thus, the two numbers are:

(x 17) (y 5)

Alternative Solutions

There are several alternative solutions to this problem, and each involves different substitution and simplification methods:

(5y - 7 22) simplifies to (5y 29), leading to (y 3), and thus (x 10). (2a - 8b 22) with (b 2a) simplifies to (22a 22), leading to (a 1) and (b 2). (n 2m) and (6m - 8n 22) simplify to (m 1) and (n 2). (6n - 8(2n) 22) simplifies to (22n 22), leading to (n 1) and (x 2).

Conclusion

In conclusion, the problem of finding two numbers where one is 7 more than twice the other and their sum is 22 can be solved through systematic substitution and simplification of equations. This example demonstrates the versatility and power of linear equations in representing and solving real-world scenarios.