Of two numbers, 4 times the smaller one is less than 3 times the larger one by 6. if the sum of the numbers is larger than 6 times their difference by 5, find the larger numbers. (do it in simpler way)

Let's break down the problem and solve it step-by-step in a simpler way.

1. **Assign Variables:**
   - Let the two numbers be \( x \) and \( y \), where \( x \) is the smaller number and \( y \) is the larger number.

2. **Set Up Equations Based on the Problem Statements:**

   - **First Statement:** "4 times the smaller one is less than 3 times the larger one by 6."
     - This translates to:  
       \[
       4x = 3y - 6
       \]

   - **Second Statement:** "The sum of the numbers is larger than 6 times their difference by 5."
     - First, the difference between \( y \) and \( x \) is \( y - x \).
     - So, 6 times their difference is \( 6(y - x) \).
     - We have:  
       \[
       x + y = 6(y - x) + 5
       \]

3. **Solve the Equations Step-by-Step:**

   Let's start by simplifying the second equation.

   \[
   x + y = 6(y - x) + 5
   \]
   Expanding and rearranging:
   \[
   x + y = 6y - 6x + 5
   \]
   Bringing like terms together:
   \[
   7x = 5y - 5
   \]

   Now, we have a system of equations:

   \[
   4x = 3y - 6
   \]
   \[
   7x = 5y - 5
   \]

4. **Solve the System of Equations:**

   - Multiply the first equation by 5 and the second equation by 3 to make the coefficients of \( y \) equal:

     \[
     20x = 15y - 30
     \]
     \[
     21x = 15y - 15
     \]

   - Subtract the first modified equation from the second:

     \[
     21x - 20x = (15y - 15) - (15y - 30)
     \]
     \[
     x = 15
     \]

   - Substitute \( x = 15 \) into the first equation to find \( y \):

     \[
     4(15) = 3y - 6
     \]
     \[
     60 = 3y - 6
     \]
     \[
     3y = 66
     \]
     \[
     y = 22
     \]

5. **Conclusion:**
   - The larger number is \( y = 22 \).

Let me know if you'd like further explanation or if you have any questions.

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### Related Questions:
1. How would the solution change if the first statement used "greater than" instead of "less than"?
2. What if both statements were equalities instead of inequalities?
3. How do we solve similar problems if given different numbers or coefficients?
4. Can we solve this problem graphically by plotting the equations?
5. How would the approach change if the sum were exactly equal to 6 times the difference?

**Tip:** When dealing with word problems, translating each statement into an equation is key.

Of two numbers, 4 times the smaller one is less than 3 times the larger one by 6. If the sum of the numbers is larger than 6 times their difference by 5, find the larger number.

Learn how to solve a system of equations involving two numbers where 4 times the smaller number is less than 3 times the larger by 6, and the sum is larger than 6 times their difference by 5. This problem covers key algebraic concepts such as linear equations and inequalities, suitable for students in Grades 8-10.

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