Simplify the given equations The given equations are:$$\frac{92}{47}k = 2s^2 + 720s$$$$\frac{92}{47}(2s^2 + 360s) = 2s^2 + 720s$$

To simplify the given equations, we will work with them one at a time. Let's start with the first equation: \[ \frac{92}{47}k = 2s^2 + 720s \] We can isolate \(k\) by multiplying both sides of the equation by \(\frac{47}{92}\): \[ k = \frac{47}{92}(2s^2 + 720s) \] Next, we will simplify the second equation: \[ \frac{92}{47}(2s^2 + 360s) = 2s^2 + 720s \] Multiply both sides by \( \frac{47}{92} \) to eliminate the fraction on the left: \[ 2s^2 + 360s = \frac{47}{92} (2s^2 + 720s) \] Now, multiply through the right side to remove the fraction: \[ 2s^2 + 360s = \frac{47 \cdot 2}{92}s^2 + \frac{47 \cdot 720}{92}s \] Calculating the coefficients: 1. For \(s^2\): \[ \frac{47 \cdot 2}{92} = \frac{94}{92} = 1.02173913 \approx 1.0217s^2 \] 2. For \(s\): \[ \frac{47 \cdot 720}{92} = \frac{33840}{92} = 367.82608696 \approx 367.8261s \] Now equating the expressions and simplifying: \[ 2s^2 + 360s = 1.0217s^2 + 367.8261s \] By moving all terms to one side to consolidate: \[ 2s^2 - 1.0217s^2 + 360s - 367.8261s = 0 \] Simplifying gives: \[ (2 - 1.0217)s^2 + (360 - 367.8261)s = 0 \] Calculating further leads us to: \[ 0.9783s^2 - 7.8261s = 0 \] Factoring out \(s\): \[ s(0.9783s - 7.8261) = 0 \] From where we get \(s = 0\) or: \[ 0.9783s = 7.8261 \quad \Rightarrow \quad s = \frac{7.8261}{0.9783} \] Calculating gives us: \[ s \approx 8 \] Thus, we derived results from both equations showing \(s \approx 8\). If you require additional coefficients' accuracy or further steps, please specify. The simplified terms indicate: 1. \( k = \frac{47}{92}(2s^2 + 720s) \) 2. \((s \text{ values from second equation give } s = 0 \text{ or } s \approx 8)\). Let me know if you need further clarification or any additional calculations! Update (2025-10-27): Let's simplify the given equations step by step. 1. **First Equation**: \[ \frac{92}{47}k = 2s^2 + 720s \] 2. **Second Equation**: \[ \frac{92}{47}(2s^2 + 360s) = 2s^2 + 720s \] We will simplify the second equation first: **Starting with the second equation**: \[ \frac{92}{47}(2s^2 + 360s) = 2s^2 + 720s \] Distributing \(\frac{92}{47}\) on the left side: \[ \frac{92}{47} \cdot 2s^2 + \frac{92}{47} \cdot 360s = 2s^2 + 720s \] \[ \frac{184}{47}s^2 + \frac{33120}{47}s = 2s^2 + 720s \] Next, we will eliminate the fractions by multiplying everything by \(47\) (the denominator): \[ 184s^2 + 33120s = 94s^2 + 33840s \] Now, we will move all terms to one side of the equation: \[ 184s^2 + 33120s - 94s^2 - 33840s = 0 \] Combine like terms: \[ (184 - 94)s^2 + (33120 - 33840)s = 0 \] This simplifies to: \[ 90s^2 - 720s = 0 \] Factoring out \(90s\): \[ 90s(s - 8) = 0 \] Thus, the solutions for \(s\) are: \[ s = 0 \quad \text{or} \quad s = 8 \] Now substituting \(s = 8\) (we'll check \(s = 0\) later) back into the first equation to find \(k\): \[ \frac{92}{47}k = 2(8)^2 + 720(8) \] Calculate \(2(8^2)\) and \(720(8)\): \[ 2(64) + 5760 = 128 + 5760 = 5888 \] Now, substitute back: \[ \frac{92}{47}k = 5888 \] Solving for \(k\): \[ k = \frac{5888 \cdot 47}{92} \] Calculating the right side: \[ \frac{5888 \cdot 47}{92} = \frac{276416}{92} = 3004 \] Thus, the values found are: \[ s = 0 \quad \text{or} \quad s = 8 \] \[ k = 3004 \] Final results: \[ k = 3004 \text{ and } s = 8 \text{ or } s = 0 \]