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{
"problem": "Find the value of $x$ that satisfies $\\frac{\\sqrt{3x+5}}{\\sqrt{6x+5}}=\\frac{\\sqrt{5}}{3}$. Express your answer as a common fraction.",
"level": "Level 4",
"type": "Algebra",
"solution": "We begin by cross multiplying and then squaring both sides \\begin{align*}\n\\frac{\\sqrt{3x+5}}{\\sqrt{6x+5}}&=\\frac{\\sqrt{5}}{3}\\\\\n3\\sqrt{3x+5}&=\\sqrt{5}\\cdot\\sqrt{6x+5}\\\\\n\\left(3\\sqrt{3x+5}\\right)^2&=\\left(\\sqrt{5}\\cdot\\sqrt{6x+5}\\right)^2\\\\\n9(3x+5) &=5(6x+5)\\\\\n20 &= 3x\\\\\nx&=\\boxed{\\frac{20}{3}}.\\\\\n\\end{align*}Checking, we see that this value of $x$ satisfies the original equation, so it is not an extraneous solution."
} |