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chords of have you ever seen the rain - SUpost

📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 $a + 3b = 5$

📌 $2a + 4b = 6$

📌 $c + 3d = 7$

📌 $2c + 4d = 8$.

📌 Solving equations 1 and 2: From 1, $a = 5 - 3b$. Substitute into 2: $2(5 - 3b) + 4b = 6 \Rightarrow 10 - 6b + 4b = 6 \Rightarrow -2b = -4 \Rightarrow b = 2$, then $a = 5 - 6 = -1$.

📌 Solving equations 3 and 4: From 3, $c = 7 - 3d$. Substitute into 4: $2(7 - 3d) + 4d = 8 \Rightarrow 14 - 6d + 4d = 8 \Rightarrow -2d = -6 \Rightarrow d = 3$, then $c = 7 - 9 = -2$.

📌 Thus, $\mathbf{M} = \begin{pmatrix} -1 & 2 \\ -2 & 3 \end{pmatrix}$. Final answer: $\boxed{\begin{pmatrix} -1 & 2 \\ -2 & 3 \end{pmatrix}}$.

📌 Question: Find all angles $z \in [0^\circ, 360^\circ]$ that satisfy $2\sin(2z) = \sqrt{3}$.

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