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 "task": "math8-linear",
 "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.",
 "system": "You are helping a grade 3-8 teacher. Produce classroom-ready material.",
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 "ran_at": "2026-10-16T08:09:17-0700",
 "output": "# 8th Grade Math Quiz: Solving Linear Equations in One Variable\n\n**Name:** ___________________________  \n**Date:** ___________________________  \n**Score:** ______ / 8\n\n**Instructions:** Solve each equation for $x$. Show all your work. For the last two problems, determine if the equation has one solution, no solution, or infinitely many solutions.\n\n---\n\n### Part 1: Basic & Distributive Property (Questions 1\u20134)\n\n**1.** Solve for $x$:  \n$$3x + 5 = 20$$\n\n**2.** Solve for $x$:  \n$$\\frac{x}{4} - 2 = 6$$\n\n**3.** Solve for $x$:  \n$$2(x - 3) = 10$$\n\n**4.** Solve for $x$:  \n$$-5(x + 2) = 15$$\n\n---\n\n### Part 2: Variables on Both Sides (Questions 5\u20136)\n\n**5.** Solve for $x$:  \n$$4x + 7 = 2x - 5$$\n\n**6.** Solve for $x$:  \n$$6x - 3 = 3x + 12$$\n\n---\n\n### Part 3: Special Cases (Questions 7\u20138)\n\n**7.** Solve for $x$:  \n$$3x + 8 = 3x - 2$$\n\n**8.** Solve for $x$:  \n$$2(x + 4) = 2x + 8$$\n\n---\n---\n\n# Answer Key & Step-by-Step Solutions\n\n### **1.** $3x + 5 = 20$\n*   **Step 1:** Subtract 5 from both sides.\n    $$3x = 15$$\n*   **Step 2:** Divide both sides by 3.\n    $$x = 5$$\n*   **Answer:** $x = 5$\n\n### **2.** $\\frac{x}{4} - 2 = 6$\n*   **Step 1:** Add 2 to both sides.\n    $$\\frac{x}{4} = 8$$\n*   **Step 2:** Multiply both sides by 4.\n    $$x = 32$$\n*   **Answer:** $x = 32$\n\n### **3.** $2(x - 3) = 10$\n*   **Step 1:** Distribute the 2 into the parentheses.\n    $$2x - 6 = 10$$\n*   **Step 2:** Add 6 to both sides.\n    $$2x = 16$$\n*   **Step 3:** Divide both sides by 2.\n    $$x = 8$$\n*   **Answer:** $x = 8$\n\n### **4.** $-5(x + 2) = 15$\n*   **Step 1:** Distribute the -5 into the parentheses.\n    $$-5x - 10 = 15$$\n*   **Step 2:** Add 10 to both sides.\n    $$-5x = 25$$\n*   **Step 3:** Divide both sides by -5.\n    $$x = -5$$\n*   **Answer:** $x = -5$\n\n### **5.** $4x + 7 = 2x - 5$\n*   **Step 1:** Subtract $2x$ from both sides to get variables on one side.\n    $$2x + 7 = -5$$\n*   **Step 2:** Subtract 7 from both sides to get constants on the other side.\n    $$2x = -12$$\n*   **Step 3:** Divide both sides by 2.\n    $$x = -6$$\n*   **Answer:** $x = -6$\n\n### **6.** $6x - 3 = 3x + 12$\n*   **Step 1:** Subtract $3x$ from both sides.\n    $$3x - 3 = 12$$\n*   **Step 2:** Add 3 to both sides.\n    $$3x = 15$$\n*   **Step 3:** Divide both sides by 3.\n    $$x = 5$$\n*   **Answer:** $x = 5$\n\n### **7.** $3x + 8 = 3x - 2$\n*   **Step 1:** Subtract $3x$ from both sides.\n    $$8 = -2$$\n*   **Step 2:** Analyze the result. The variable $x$ has disappeared, and the statement $8 = -2$ is **false**.\n*   **Answer:** **No Solution**\n\n### **8.** $2(x + 4) = 2x + 8$\n*   **Step 1:** Distribute the 2 on the left side.\n    $$2x + 8 = 2x + 8$$\n*   **Step 2:** Subtract $2x$ from both sides.\n    $$8 = 8$$\n*   **Step 3:** Analyze the result. The variable $x$ has disappeared, and the statement $8 = 8$ is **true**. This means the equation is true for any value of $x$.\n*   **Answer:** **Infinitely Many Solutions**",
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 "retry_note": "ran with a 1,500-token budget: Groq's free tier now rejects Qwen3.8 27B requests whose expected output exceeds its 1,000 output-tokens-per-minute limit"
}