Solving a System of Equations to Find Two Numbers
In this article, we will solve a system of linear equations to find two numbers. The problem is given in the form of two equations, and we will walk through the steps to find these numbers. This process is a fundamental concept in algebra, commonly used in solving various real-world problems. This article aims to explain the step-by-step process in a clear and detailed manner.
Problem Statement
Let's denote the two numbers as (x) (the smaller number) and (y) (the larger number). The problem provides two equations based on the given conditions:
Equation 1
The sum of the two numbers is sixteen.
[x y 16]Equation 2
The difference between four times the smaller number and two is two more than twice the larger number.
[4x - 2 2y 2]Solution Step-by-Step
Step 1: Rearrange the Equations
From the first equation, we can express (y) in terms of (x).
[y 16 - x]Step 2: Substitute into the Second Equation
Now, substitute (y 16 - x) into the second equation.
[4x - 2 2(16 - x) 2]Step 3: Simplify the Equation
Expand and simplify the equation.
[4x - 2 32 - 2x 2] [4x - 2 34 - 2x]Step 4: Combine Like Terms
Add (2x) to both sides of the equation to combine like terms.
[6x - 2 34]Add 2 to both sides.
[6x 36]Step 5: Solve for (x)
Divide by 6 to solve for (x).
[x 6]Step 6: Find (y)
Substitute (x 6) back into the equation for (y).
[y 16 - x 16 - 6 10]Final Result
The two numbers are:
(6) and (10)
Thus, the smaller number is (6) and the larger number is (10).
Alternative Method
Another approach to solve the problem is to use the following set of equations:
The sum of the two numbers is sixteen:
[x y 16]The other condition given is:
[4x - 2 2y 2]Which simplifies to:
[4x - 2y 4]Now, we can solve the system of equations:
Step 1: Express (x) in terms of (y)
[x y 4]Step 2: Substitute into the first equation
[(y 4) y 16] [2y 4 16] [2y 12] [y 6]Step 3: Find (x)
[x y 4 6 4 10]Verification
To verify the solution, substitute (x 6) and (y 10) back into the original equations.
Equation 1
[6 10 16]This is true.
Equation 2
[4(6) - 2 2(10) 2] [24 - 2 20 2] [22 22]This is also true.
Conclusion
Thus, the correct solution to the given problem is:
(6) and (10)
This solution provides a clear illustration of how to solve a system of equations in algebra. By solving such problems, one can enhance their understanding of algebraic concepts and apply them to various real-life situations.
Keywords
Solving equations, system of linear equations, algebra problems