\( x = \pm 3 \): \( 9(9) = 81 \), \( 144 - 81 = 63 \), \( y^2 =

["Understanding the Equation: ( x = \pm 3 ), ( 9x^2 = 81 ), and the Next Step: ( y^2 = ? )", "When solving equations involving integers, clarity and step-by-step reasoning are essential. One commonly explored example involves the equation:", "[ x = \pm 3 ]\nwith the fact that:\n[ 9x^2 = 81 ]\nfollowed by the calculation:\n[ 144 - 81 = 63 ]\nand finally, the expression to determine:\n[ y^2 = ? ]", "Let’s unpack each step to understand how this equation unfolds and what value ( y^2 ) represents.", "---", "### Step 1: Confirming ( x = \pm 3 ) from ( 9x^2 = 81 )", "Start with the equation:", "[ 9x^2 = 81 ]", "To isolate ( x^2 ), divide both sides by 9:", "[\nx^2 = \frac{81}{9} = 9\n]", "Taking the square root of both sides:", "[\nx = \pm \sqrt{9} = \pm 3\n]", "This confirms that ( x = 3 ) or ( x = -3 ), consistent with the given ( x = \pm 3 ).", "---", "### Step 2: Evaluating ( 144 - 81 = 63 )", "We are told:", "[ 144 - 81 = 63 ]", "This supports the value of ( x^2 = 9 ), since:", "[\n144 - 81 = 9x^2 = 81\n]", "So, the expression ( 144 - 81 ) gives us 63 — which aligns with the earlier substitution and confirms the truth of ( x^2 = 9 ).", "---", "### Step 3: Solving ( y^2 = ? )", "The next stage asks us to compute:", "[ y^2 = ? ]", "To identify ( y^2 ), we must determine what value ( y ) represents in this context. While the problem does not specify ( y ) directly, a common interpretation—especially in algebraic or geometric problems—is that this relation follows from ( x ) values used in forming an equation.", "Given the prior expression ( 144 - 81 = 63 ), which equals ( 9x^2 ), and knowing ( x^2 = 9 ), we may consider that ( y^2 ) represents a derived quantity from the same relationships. One logical path is:", "[\ny^2 = 144 - 9x^2\n]", "Since ( 9x^2 = 81 ):", "[\ny^2 = 144 - 81 = 63\n]", "Thus,", "[\ny^2 = 63\n]", "---", "### Conclusion: The Value and Significance", "In summary, starting from ( x = \pm 3 ), we derived ( x^2 = 9 ). Using ( 144 - 81 = 63 ), we identify:", "[\ny^2 = 63\n]", "This value arises naturally from the difference between 144 and the prior result from ( x^2 ), reinforcing how algebraic identities can guide problem-solving. Whether in quadratic equations, geometry, or number theory, recognizing these relationships helps simplify complex problems.", "---", "### Key Takeaways:", "- ( x = \pm 3 ) leads to ( x^2 = 9 ).\n- ( 144 - 81 = 63 ) confirms ( x^2 = 9 ) in context.\n- ( y^2 = 144 - 9x^2 = 63 ) follows logically.\n- Understanding stepwise derivations improves problem-solving accuracy and clarity.", "If you're exploring equations in middle or high school math, mastering such relationships strengthens your foundation for algebra, calculus, and beyond. Keep practicing — every equation tells a story!"]