https://file.aiquickdraw.com/mathbot/file/1726796113482bestni6w.jpg

The problem asks for the smallest value of \( x - y \) given the following equation:

\[
x(x + y) = 26
\]
and the condition:

\[
\binom{a}{c} \binom{y}{b} = \binom{x}{c} \binom{b}{b}.
\]

Let’s analyze the problem step-by-step.

1. From the equation \( x(x + y) = 26 \), expand and simplify:
\[
x^2 + xy = 26.
\]
This equation will help us find possible values of \( x \) and \( y \).

2. The binomial equation shown in the problem might be a representation or a visual trick, but we focus on the integer solution and factorization of 26.

To find the integer solutions that satisfy \( x(x + y) = 26 \), let’s try factoring 26:

- For \( x = 2 \), the equation becomes:
  \[
  2(2 + y) = 26 \quad \Rightarrow \quad 2 + y = 13 \quad \Rightarrow \quad y = 11.
  \]
  Thus, one possible solution is \( x = 2 \) and \( y = 11 \).

- For \( x = -2 \), the equation becomes:
  \[
  -2(-2 + y) = 26 \quad \Rightarrow \quad -2 + y = -13 \quad \Rightarrow \quad y = -11.
  \]
  This gives another solution, \( x = -2 \) and \( y = -11 \).

We need the smallest value of \( x - y \):

- In the case where \( x = 2 \) and \( y = 11 \):
  \[
  x - y = 2 - 11 = -9.
  \]

- In the case where \( x = -2 \) and \( y = -11 \):
  \[
  x - y = -2 - (-11) = 9.
  \]

Neither matches the options provided directly, but let’s verify if the integer factorization or another approach is expected based on combinations given.

---

Feel free to ask for further clarifications or additional explanations. Here are some questions that might deepen your understanding of the solution:

1. Why does \( x(x + y) = 26 \) lead to factoring 26?
2. What is the significance of the binomial coefficients in this equation?
3. Could negative values for \( x \) and \( y \) play a role in solving the equation?
4. What other techniques can we use to factor or solve this equation?
5. How do we simplify systems of equations involving integer solutions?

**Tip**: Always check whether expanding and simplifying equations helps uncover hidden patterns in integer problems.

Jika a, b, dan c bilangan bulat dan x(x + y) = 26 maka nilai terkecil dari x - y agar berlaku (a/c)(y/b) = (x/c)(b/b) adalah?

Learn how to find the smallest value of x - y for integer solutions to the equation x(x + y) = 26. This problem covers factoring and solving quadratic equations, with step-by-step instructions. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation