Solving Algebraic Equations: A Problem with Numbers

Algebra is a fundamental part of mathematics that focuses on manipulating symbols and formulas to solve equations and find unknown values. This article explores a basic problem involving algebraic equations, emphasizing the steps and reasoning involved in solving it. The problem we will tackle involves two numbers, where the larger number is 10 more than the smaller. Additionally, we are given that the sum of twice the smaller number and three times the larger number equals 55. Let's break down the process of finding the values of these two numbers.

Defining the Variables

We start by defining the two numbers. Let the smaller number be represented by ( x ). Since the larger number is 10 more than the smaller, we can express it as ( x 10 ).

Setting Up the Equation

The problem statement gives us the condition that the sum of twice the smaller number and three times the larger number equals 55. We can set this up as an equation:

[ 2x 3(x 10) 55 ]

Now, let's simplify this equation step by step to find ( x ).

Simplifying the Equation

First, distribute the 3 to both ( x ) and 10:

[ 2x 3x 30 55 ]

Next, combine the like terms ( 2x ) and ( 3x ):

[ 5x 30 55 ]

Then, isolate the term with ( x ) by subtracting 30 from both sides of the equation:

[ 5x 25 ]

Finally, solve for ( x ) by dividing both sides by 5:

[ x 5 ]

Finding the Larger Number

Now that we know the value of the smaller number ( x ), we can find the larger number. Since the larger number is ( x 10 ), we have:

[ x 10 5 10 15 ]

Therefore, the two numbers are:

Smaller number: 5 Larger number: 15

Additional Solutions from Different Approaches

The problem can be solved in several ways, and here are a few alternative methods:

Method 1: Using Substitution and Simplification

One approach starts with the equation:

2x 3(x - 10) 55

Distributing and simplifying:

2x 3x - 30 55

5x - 30 55

5x 85

x 17

This solution does not match the previous one, indicating an error in the setup or calculation.

Method 2: Solving Directly

Starting with the correct setup:

2x 3(x 10) 55

Simplifies to:

2x 3x 30 55

5x 25

x 5

And verify:

2(5) 3(15) 55

10 45 55

Conclusion

The correct solution to the problem is that the smaller number is 5 and the larger number is 15. Understanding and solving such problems is crucial for developing algebraic reasoning and problem-solving skills. Whether through substitution, simplification, or direct calculation, the core principle remains the same: isolate and solve for the unknown variable to find the solution.