For what exact value of x do the two linear expressions 15 − 7x and 15x + 7 become equal? Solve the equation and give x as a simplified fraction.

Introduction / Context:
This is a basic linear-equation problem. When two expressions are said to be equal, we set them equal and solve for the variable. Such questions test algebraic manipulation: moving x terms to one side, constants to the other, and simplifying correctly. Since the expressions are linear in x, the solution is unique and can be expressed exactly as a fraction.

Given Data / Assumptions:

• Expression 1: 15 - 7x • Expression 2: 15x + 7 • Condition: 15 - 7x = 15x + 7 • Required: x in simplest fraction form

Concept / Approach:
Set the expressions equal and solve: 15 - 7x = 15x + 7. Collect x terms on one side and constants on the other. Then divide by the coefficient of x to isolate x. Keep all steps exact to avoid sign mistakes.

Step-by-Step Solution:
1) Set the expressions equal: 15 - 7x = 15x + 7 2) Move x terms to one side by adding 7x to both sides: 15 = 22x + 7 3) Move constants by subtracting 7 from both sides: 8 = 22x 4) Divide both sides by 22: x = 8/22 5) Simplify the fraction by dividing numerator and denominator by 2: x = 4/11

Verification / Alternative check:
Substitute x = 4/11: 15 - 7x = 15 - 28/11 = (165/11 - 28/11) = 137/11. 15x + 7 = 60/11 + 77/11 = 137/11. Both sides match exactly, confirming the solution.

Why Other Options Are Wrong:
• -4/11 would flip the sign and make the two expressions differ. • 11/4 and -11/4 are reciprocals/negatives and do not satisfy the original equation. • 7/22 is not obtained from the correct isolation of x and fails substitution.

Common Pitfalls:
• Moving -7x across the equals sign without changing sign properly. • Forgetting to simplify 8/22 to 4/11.

Final Answer:
4/11