In basic algebra, solve for x in the linear equation in one variable: (9 − 3x) − (17x − 10) = −1. What is the value of x that satisfies this equation exactly?

Introduction / Context:
This question tests your ability to solve a basic linear equation in one variable that involves brackets and negative signs. Such equations are extremely common in school algebra, aptitude tests, and many real life applications where you need to isolate an unknown. The main skills required are expanding brackets correctly, combining like terms, and performing the same operation on both sides of the equation to preserve equality.

Given Data / Assumptions:

• The equation is (9 − 3x) − (17x − 10) = −1.
• x is a real number.
• Standard rules of algebra apply, including distribution over brackets and the equality principle.
• We seek the exact value of x that satisfies this equation.

Concept / Approach:
To solve this equation, we first remove brackets using distribution. The key idea is that when you subtract a bracket, you change the sign of every term inside that bracket. After expansion, we collect all x terms together and all constant terms together. Finally, we isolate x by performing simple addition, subtraction, or division steps. The equation is linear, so it will have at most one solution for x.

Step-by-Step Solution:
1) Start with (9 − 3x) − (17x − 10) = −1. 2) Expand the first part: 9 − 3x remains as it is; there is no leading coefficient to distribute. 3) Subtract the second bracket by changing every sign inside it: −(17x − 10) becomes −17x + 10. 4) Combine terms on the left side: (9 − 3x) − 17x + 10 = 9 + 10 − 3x − 17x = 19 − 20x. 5) The equation becomes 19 − 20x = −1. 6) Subtract 19 from both sides to move constants to the right: −20x = −1 − 19 = −20. 7) Divide both sides by −20: x = (−20) / (−20) = 1.

Verification / Alternative check:
Substitute x = 1 back into the original equation. Compute 9 − 3x = 9 − 3 * 1 = 6. Next, compute 17x − 10 = 17 * 1 − 10 = 7. The left side becomes (9 − 3x) − (17x − 10) = 6 − 7 = −1, which matches the right side exactly. This confirms that x = 1 is indeed the correct solution and that there is no arithmetic mistake in the steps above.

Why Other Options Are Wrong:
Option b (−1) would give (9 − 3(−1)) − (17(−1) − 10) = 12 − (−17 − 10) which does not simplify to −1. Option c (9/10), option d (−9/10), and option e (2) likewise produce left side values different from −1 when substituted into the equation. Only x = 1 satisfies the equality exactly.

Common Pitfalls:
A common mistake is to forget to change the sign of both terms in the second bracket when subtracting (17x − 10). Some learners also incorrectly combine constants or mishandle negative numbers when moving terms across the equality sign. Writing each step carefully, especially the sign changes when removing brackets, helps avoid these errors and leads quickly to the correct value of x.

Final Answer:
The linear equation is satisfied when x equals 1.