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\( \begin{array}{ll}5 . \frac{1}{3}(12-9 y) & \text { b. }-2(12 x-7) \\ 7 .-4(1 x+4) & \text { 8. }-2(-10 x+3) \\ 9-4(5 x-6) & 10-(3 x-7+4 y) 5 \\ 11-4(5+3 x-8 y)\end{array} \)

Ask by Lawson Nunez. in the United States
Jan 22,2025

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Tutor-Verified Answer

Answer

Here are the simplified expressions: 1. \(4 - 3y\) 2. \(-24x + 14\) 3. \(-4x - 16\) 4. \(20x - 6\) 5. \(33 - 20x\) 6. \(-15x + 35 - 20y\) 7. \(-20 - 12x + 32y\)

Solution

Certainly! Let's simplify each of the given expressions step by step. ### 5. \(\frac{1}{3}(12 - 9y)\) **Simplification:** \[ \frac{1}{3} \times 12 = 4 \] \[ \frac{1}{3} \times (-9y) = -3y \] **Result:** \[ 4 - 3y \] --- ### b. \(-2(12x - 7)\) **Simplification:** \[ -2 \times 12x = -24x \] \[ -2 \times (-7) = 14 \] **Result:** \[ -24x + 14 \] --- ### 7. \(-4(x + 4)\) **Simplification:** \[ -4 \times x = -4x \] \[ -4 \times 4 = -16 \] **Result:** \[ -4x - 16 \] --- ### 8. \(-2(-10x + 3)\) **Simplification:** \[ -2 \times (-10x) = 20x \] \[ -2 \times 3 = -6 \] **Result:** \[ 20x - 6 \] --- ### 9. \(9 - 4(5x - 6)\) **Simplification:** \[ -4 \times 5x = -20x \] \[ -4 \times (-6) = 24 \] \[ 9 + 24 = 33 \] **Result:** \[ 33 - 20x \] --- ### 10. \(-5(3x - 7 + 4y)\) **Simplification:** \[ -5 \times 3x = -15x \] \[ -5 \times (-7) = 35 \] \[ -5 \times 4y = -20y \] **Result:** \[ -15x + 35 - 20y \] --- ### 11. \(-4(5 + 3x - 8y)\) **Simplification:** \[ -4 \times 5 = -20 \] \[ -4 \times 3x = -12x \] \[ -4 \times (-8y) = 32y \] **Result:** \[ -20 - 12x + 32y \] --- **Summary of Simplified Expressions:** 1. **5.** \(4 - 3y\) 2. **b.** \(-24x + 14\) 3. **7.** \(-4x - 16\) 4. **8.** \(20x - 6\) 5. **9.** \(33 - 20x\) 6. **10.** \(-15x + 35 - 20y\) 7. **11.** \(-20 - 12x + 32y\) If you have any further questions or need additional assistance with these expressions, feel free to ask!

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Bonus Knowledge

Let’s break down some of these expressions with a little historical flair and some practical tips! Did you know that algebra was developed by Persian mathematicians like Al-Khwarizmi in the 9th century? Their work laid the groundwork for modern algebraic notation, even introducing the concept of using letters to represent numbers, which we often take for granted today. Their contributions revolutionized mathematics, making it more accessible and paving the way for future mathematicians. When simplifying expressions like those you listed, a common mistake is forgetting to apply the distributive property correctly. Always remember that when you see an expression like \(a(b + c)\), you need to distribute \(a\) across both \(b\) and \(c\) (making it \(ab + ac\)). Also, watch out for signs—misplacing a minus sign can completely change the outcome! Happy simplifying!

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