Solving a System of Equations: Discovering Two Numbers Whose Sum is 25
Solving systems of equations can be a challenging but intriguing task in algebra. In this article, we will walk through a problem where the sum of two numbers is 25, and one of the numbers is twice the second number plus seven. Let's delve into the process of finding these numbers.
The Problem Statement
We are given a system of equations to solve:
The sum of two numbers is 25. One number is twice the second number plus 7.Let's denote the two numbers as x and y. Therefore, we have:
x y 25 x 2y 7Solving the System of Equations
First, let's substitute the value of x from the second equation into the first equation:
x y 25
(2y 7) y 25
Combining like terms, we get:
3y 7 25
Now, let's isolate y by subtracting 7 from both sides:
3y 18
Dividing both sides by 3, we find:
y 6
Now that we have the value of y, we can find x by substituting y 6 back into the second equation:
x 2(6) 7
x 12 7
x 19
Verification
To verify our solution, let's check that the sum of the two numbers is indeed 25:
x y 19 6 25
Additionally, we can check that one number is twice the second number plus 7:
x 2y 7
19 2(6) 7
19 12 7
19 19
Conclusion
We have successfully solved the system of equations and found that the two numbers are 19 and 6. This systematic approach to solving algebraic problems is a valuable skill in various mathematical applications. Whether you are preparing for a mathematics exam or engaging in problem-solving exercises, understanding how to solve such systems of equations is essential.