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 "prompt": "Write an 8-question quiz for 8th graders on solving linear equations in one variable, including equations with variables on both sides, distributing, and one equation with no solution and one with infinitely many solutions. Provide a step-by-step answer key.",
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 "ran_at": "2026-10-16T07:58:20-0700",
 "output": "## 8\u2011th Grade Quiz \u2013 Solving Linear Equations in One Variable  \n\n**Name:** _______________________\u2003\u2003**Date:** _______________________  \n\nSolve each equation. Write the value of the variable in the space provided. Show your work on the back of the paper (or on the extra lines below each problem) \u2013 you will need the steps for the answer key.\n\n---\n\n### Questions  \n\n| # | Equation | Solution (blank) |\n|---|----------|------------------|\n| 1 | \\(3x + 5 = 20\\) | __________________ |\n| 2 | \\(2x - 7 = x + 4\\) | __________________ |\n| 3 | \\(4(2x - 3) = 28\\) | __________________ |\n| 4 | \\(5x + 2 = 3(x + 6)\\) | __________________ |\n| 5 | \\(\\displaystyle \\frac{3x}{4} - 2 = 5\\) | __________________ |\n| 6 | \\(6x + 2 = 6x - 5\\) | __________________ |\n| 7 | \\(4(x - 1) = 4x - 4\\) | __________________ |\n| 8 | \\(7 - 2(3x - 1) = 5x + 9\\) | __________________ |\n\n---\n\n### Answer Key \u2013 Step\u2011by\u2011Step Solutions  \n\n#### 1. \\(3x + 5 = 20\\)\n\n1. Subtract 5 from both sides:\u2003\\(3x = 20 - 5\\) \u2192 \\(3x = 15\\)  \n2. Divide both sides by 3:\u2003\\(x = \\dfrac{15}{3}\\) \u2192 **\\(x = 5\\)**  \n\n---\n\n#### 2. \\(2x - 7 = x + 4\\)\n\n1. Subtract \\(x\\) from both sides:\u2003\\(2x - x - 7 = 4\\) \u2192 \\(x - 7 = 4\\)  \n2. Add 7 to both sides:\u2003\\(x = 4 + 7\\) \u2192 **\\(x = 11\\)**  \n\n---\n\n#### 3. \\(4(2x - 3) = 28\\)\n\n1. Distribute the 4:\u2003\\(8x - 12 = 28\\)  \n2. Add 12 to both sides:\u2003\\(8x = 28 + 12\\) \u2192 \\(8x = 40\\)  \n3. Divide by 8:\u2003\\(x = \\dfrac{40}{8}\\) \u2192 **\\(x = 5\\)**  \n\n---\n\n#### 4. \\(5x + 2 = 3(x + 6)\\)\n\n1. Distribute the 3 on the right:\u2003\\(5x + 2 = 3x + 18\\)  \n2. Subtract \\(3x\\) from both sides:\u2003\\(5x - 3x + 2 = 18\\) \u2192 \\(2x + 2 = 18\\)  \n3. Subtract 2 from both sides:\u2003\\(2x = 16\\)  \n4. Divide by 2:\u2003\\(x = \\dfrac{16}{2}\\) \u2192 **\\(x = 8\\)**  \n\n---\n\n#### 5. \\(\\displaystyle \\frac{3x}{4} - 2 = 5\\)\n\n1. Add 2 to both sides:\u2003\\(\\frac{3x}{4} = 7\\)  \n2. Multiply both sides by 4 to clear the denominator:\u2003\\(3x = 28\\)  \n3. Divide by 3:\u2003\\(x = \\dfrac{28}{3}\\) \u2192 **\\(x = \\frac{28}{3}\\) (or 9\u202f\u2153)**  \n\n---\n\n#### 6. \\(6x + 2 = 6x - 5\\)\u2003*(No solution)*  \n\n1. Subtract \\(6x\\) from both sides:\u2003\\(2 = -5\\)  \n2. The statement \\(2 = -5\\) is false, so **there is no value of \\(x\\) that makes the equation true**.  \n   \u2192 **No solution**  \n\n---\n\n#### 7. \\(4(x - 1) = 4x - 4\\)\u2003*(Infinitely many solutions)*  \n\n1. Distribute the 4 on the left:\u2003\\(4x - 4 = 4x - 4\\)  \n2. Subtract \\(4x\\) from both sides:\u2003\\(-4 = -4\\)  \n3. The resulting statement is always true, regardless of the value of \\(x\\).  \n   \u2192 **Infinitely many solutions** (the equation is an identity).  \n\n---\n\n#### 8. \\(7 - 2(3x - 1) = 5x + 9\\)\n\n1. Distribute the \\(-2\\):\u2003\\(7 - 6x + 2 = 5x + 9\\) \u2192 combine constants: \\(9 - 6x = 5x + 9\\)  \n2. Add \\(6x\\) to both sides:\u2003\\(9 = 11x + 9\\)  \n3. Subtract 9 from both sides:\u2003\\(0 = 11x\\)  \n4. Divide by 11:\u2003\\(x = 0\\)  \n\n**Answer:** **\\(x = 0\\)**  \n\n---\n\n### Quick Review of Key Concepts  \n\n| Concept | What to Do |\n|---------|------------|\n| **Isolate the variable** | Move all terms with the variable to one side and constants to the other. |\n| **Distribute** | Multiply the outside number by each term inside the parentheses. |\n| **Combine like terms** | Add or subtract coefficients of the same variable (or constants). |\n| **No solution** | After simplifying, you get a false statement (e.g.,\u202f\\(2 = -5\\)). |\n| **Infinitely many solutions** | After simplifying, you get a true statement (e.g.,\u202f\\(0 = 0\\)). |\n\nGood luck! \ud83c\udf89",
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