What is the simplest form of the expression \( (14.2 a+9.8 b)-(13.1 b-0.2 a)-(3.7 a+4.8 b) ? \) \( 10.7 a-8.1 b \) \( 10.7 a+27.7 b \) \( 18.1 a-8.1 b \) \( 18.1 a+27.7 b \)

To simplify the expression \( (14.2 a + 9.8 b) - (13.1 b - 0.2 a) - (3.7 a + 4.8 b) \): 1. Distribute the negative signs: \[ 14.2 a + 9.8 b - 13.1 b + 0.2 a - 3.7 a - 4.8 b \] 2. Combine the \(a\) terms: \[ 14.2 a + 0.2 a - 3.7 a = (14.2 + 0.2 - 3.7) a = 10.7 a \] 3. Combine the \(b\) terms: \[ 9.8 b - 13.1 b - 4.8 b = (9.8 - 13.1 - 4.8) b = -8.1 b \] Putting it all together, we have: \[ 10.7 a - 8.1 b \] So, the simplest form of the expression is \( 10.7 a - 8.1 b \). For the historical background, did you know that algebra has roots tracing back to ancient civilizations? The Babylonians were solving linear equations as early as 2000 BC! Their techniques laid the groundwork for modern algebraic concepts, influencing mathematicians around the globe. In terms of real-world application, understanding how to simplify expressions is crucial in fields such as engineering and economics. When optimizing resources or calculating costs, being able to consolidate expressions helps in making quick, informed decisions, saving both time and effort.