- The expressions are simplified by applying the distributive property and combining like terms.
- For equations, the variable \(x\) is isolated using inverse algebraic operations.
- The results are: 1: \(4x+12\), 2: \(x=11, 2.5\), 3: \(4x^5 + x^2 + 7x + 5\), 4: \(-3x-1\), 5: \(7x-2\), 6: \(3x-7\) and \(\frac{-97x+64}{4x-4}\), 7: \(36x-5\), 8: \(13x+14\), 9: \(x=-\frac{20}{3}\), 10: \(22x-13\), 11: \(4x-52\), 12: \(-9x-42\).
- Final answers are summarized individually for clarity.
Explanation
- Systematic Approach to Algebraic Simplification and Equation Solving
To simplify algebraic expressions and solve equations, we apply the distributive property \(a(b+c) = ab + ac\) to remove parentheses, group like terms, and perform inverse operations to isolate the variable \(x\). We systematically address each item provided in the list.
- Simplification and Solution Results
- \(4(x+3) = 4x + 12\)
- For \(5x = 55\), dividing by 5 gives \(x = 11\). For \(2x = 5\), dividing by 2 gives \(x = 2.5\).
- Combining like terms: \(4x^5 + x^2 + 7x + (2 + 3) = 4x^5 + x^2 + 7x + 5\).
- \(9 - 5x + 2x - 9 - 1 = (9 - 9 - 1) + (-5x + 2x) = -1 - 3x\).
- \(6x - 2 + x = 7x - 2\).
- First part: \(8 + 3x - 15 = 3x - 7\). Second part: \(\frac{3x - 36}{4x - 4} - 25 = \frac{3x - 36 - 100(x - 1)}{4x - 4} = \frac{3x - 36 - 100x + 100}{4x - 4} = \frac{-97x + 64}{4x - 4}\).
- \(28x - 14 + 8x + 9 = 36x - 5\).
- \(4 + 15x + 10 - 2x = 13x + 14\).
- \(-4x + 36 - 3 + 7x = 13 \implies 3x + 33 = 13 \implies 3x = -20 \implies x = -\frac{20}{3}\).
- \(18x + 27 + 4x - 40 = 22x - 13\).
- \(6x - 42 - 2x - 10 = 4x - 52\).
- \(8 - 12x - 36 + 2x - 14 + x = -9x - 42\).
Examples
Algebraic simplification is essential in fields like engineering and physics. For instance, when designing a circuit, you might arrive at a complex expression for total resistance. Simplifying this expression allows you to understand how changing components, like resistors, directly impacts the circuit's overall performance. Being able to solve equations and simplify expressions is a foundational skill for modeling and optimizing systems in everyday technology and science applications.