Question

directions: simplify each expression by combining like terms.
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directions: simplify each expression using the distributive property.
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directions: simplify each expression completely.
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1. laura bought y yards of fabric on saturday. the next day, she went back and bought 3 more yards than she did on saturday. if each yard of fabric costs $5, write an expression in simplest form for the total amount of money she spent on both days.
2. jonas is a runner. he ran m miles last week. his goal this week was to run one mile less than twice the miles he ran last week. if he tripled his goal, write an expression in simplest form to show how many more miles he ran this week compared to last.

Answer:

Let's solve these problems one by one. We'll start with the first section (combining like terms), then the distributive property, then the mixed problems, and finally the word problems.

Section 1: Combining Like Terms

Problem 1: \(-x + 9 + 7x\)

• Combine like terms (\(-x\) and \(7x\)): \((-x + 7x) + 9 = 6x + 9\)

Problem 2: \(5b - 14 - 1 - 3b\)

• Combine like terms (\(5b\) and \(-3b\); \(-14\) and \(-1\)): \((5b - 3b) + (-14 - 1) = 2b - 15\)

Problem 3: \(-10r - 2r + 4 - 7r + 3\)

• Combine like terms (\(-10r\), \(-2r\), \(-7r\); \(4\) and \(3\)): \((-10r - 2r - 7r) + (4 + 3) = -19r + 7\)

Problem 4: \(8m - 20 - 6m - m + 7\)

• Combine like terms (\(8m\), \(-6m\), \(-m\); \(-20\) and \(7\)): \((8m - 6m - m) + (-20 + 7) = m - 13\)

Problem 5: \(2b + a - 4a - 2 - 12b\)

• Combine like terms (\(2b\) and \(-12b\); \(a\) and \(-4a\)): \((2b - 12b) + (a - 4a) - 2 = -10b - 3a - 2\)

Problem 6: \(\frac{1}{3}c + \frac{5}{6}c + c\)

• Convert to sixths to combine: \(\frac{2}{6}c + \frac{5}{6}c + \frac{6}{6}c = \frac{2 + 5 + 6}{6}c = \frac{13}{6}c\) or \(2\frac{1}{6}c\)

Section 2: Distributive Property

Problem 7: \(8(x + 2)\)

• Distribute 8: \(8x + 16\)

Problem 8: \(-3(w - 7)\)

• Distribute -3: \(-3w + 21\)

Problem 9: \(5(7 - n)\)

• Distribute 5: \(35 - 5n\)

Problem 10: \(-4(2r + 9s - 1)\)

• Distribute -4: \(-8r - 36s + 4\)

Section 3: Simplify Completely (Distributive + Combine Like Terms)

Problem 11: \(4(x - 5) + 2x\)

• Distribute 4: \(4x - 20 + 2x\)
• Combine like terms: \(6x - 20\)

Problem 12: \(-5 + 4(3 + p)\)

• Distribute 4: \(-5 + 12 + 4p\)
• Combine like terms: \(7 + 4p\)

Problem 13: \(20 + 7(a - 2)\)

• Distribute 7: \(20 + 7a - 14\)
• Combine like terms: \(6 + 7a\)

Problem 14: \(-2(k + 4) + k + 5\)

• Distribute -2: \(-2k - 8 + k + 5\)
• Combine like terms: \(-k - 3\)

Problem 15: \(3(8 + 5y) - 2\)

• Distribute 3: \(24 + 15y - 2\)
• Combine like terms: \(22 + 15y\)

Problem 16: \(6(7m - 1) - 10m\)

• Distribute 6: \(42m - 6 - 10m\)
• Combine like terms: \(32m - 6\)

Problem 17: \(13c - 11 - 5(c - 4)\)

• Distribute -5: \(13c - 11 - 5c + 20\)
• Combine like terms: \(8c + 9\)

Problem 18: \(18z + 2(4 - 3z) + 1\)

• Distribute 2: \(18z + 8 - 6z + 1\)
• Combine like terms: \(12z + 9\)

Problem 19: \(20 - 4 + 4(3d - 7)\)

• Simplify 20 - 4: \(16 + 4(3d - 7)\)
• Distribute 4: \(16 + 12d - 28\)
• Combine like terms: \(12d - 12\)

Problem 20: \(\frac{2}{3}(24k - 6) - 2k\)

• Distribute \(\frac{2}{3}\): \(\frac{2}{3} \cdot 24k - \frac{2}{3} \cdot 6 - 2k = 16k - 4 - 2k\)
• Combine like terms: \(14k - 4\)

Problem 21: \(7(2n - 10) + 2(4n - 1)\)

• Distribute 7 and 2: \(14n - 70 + 8n - 2\)
• Combine like terms: \(22n - 72\)

Problem 22: \(10 + 3(5a - 2b - 3) - 4b\)

• Distribute 3: \(10 + 15a - 6b - 9 - 4b\)
• Combine like terms: \(1 + 15a - 10b\)

Section 4: Word Problems

Problem 23: Laura's Fabric Cost

• Saturday: \(y\) yards, cost \(5y\) dollars.
• Sunday: \(y + 3\) yards, cost \(5(y + 3)\) dollars.
• Total cost: \(5y + 5(y + 3)\)
• Distribute and combine: \(5y + 5y + 15 = 10y + 15\) dollars.

Problem 24: Jonas's Running

• Last week: \(m\) miles.
• Goal this week: \(2m - 1\) miles.
• Tripled his goal: \(3(2m - 1) = 6m - 3\) miles.
• Difference from last week: \((6m - 3) - m = 5m - 3\) miles.

Final Answers (Selected Problems)

Let's list the answers for a few key problems:

1. \(-x + 9 + 7x = \boldsymbol{6x + 9}\)
2. \(5b - 14 - 1 - 3b = \boldsymbol{2b - 15}\)
3. \(-10r - 2r + 4 - 7r + 3 = \boldsymbol{-19r + 7}\)
4. \(8m - 20 - 6m - m + 7 = \boldsymbol{m - 13}\)
5. \(2b + a - 4a - 2 - 12b = \boldsymbol{-10b - 3a - 2}\)
6. \(\frac{1}{3}c + \frac{5}{6}c + c = \boldsymbol{\frac{13}{6}c}\) (or \(2\frac{1}{6}c\))

1. \(8(x + 2) = \boldsymbol{8x + 16}\)
2. \(-3(w - 7) = \boldsymbol{-3w + 21}\)
3. \(5(7 - n) = \boldsymbol{35 - 5n}\)
4. \(-4(2r + 9s - 1) = \boldsymbol{-8r - 36s + 4}\)

1. \(4(x - 5) + 2x = \boldsymbol{6x - 20}\)
2. \(-5 + 4(3 + p) = \boldsymbol{7 + 4p}\)
3. \(20 + 7(a - 2) = \boldsymbol{7a + 6}\) (or \(6 + 7a\))
4. \(-2(k + 4) + k + 5 = \boldsymbol{-k - 3}\)
5. \(3(8 + 5y) - 2 = \boldsymbol{15y + 22}\) (or \(22 + 15y\))
6. \(6(7m - 1) - 10m = \boldsymbol{32m - 6}\)
7. \(13c - 11 - 5(c - 4) = \boldsymbol{8c + 9}\)
8. \(18z + 2(4 - 3z) + 1 = \boldsymbol{12z + 9}\)
9. \(20 - 4 + 4(3d - 7) = \boldsymbol{12d - 12}\)
10. \(\frac{2}{3}(24k - 6) - 2k = \boldsymbol{14k - 4}\)
11. \(7(2n - 10) + 2(4n - 1) = \boldsymbol{22n - 72}\)
12. \(10 + 3(5a - 2b - 3) - 4b = \boldsymbol{15a - 10b + 1}\) (or \(1 + 15a - 10b\))

1. Total cost: \(\boldsymbol{10y + 15}\) dollars.
2. Extra miles: \(\boldsymbol{5m - 3}\) miles.