Despejar variables - Ecuaciones. Tener en cuenta: las reglas o principios. I. Despejar \( x \) ó y en cada caso teniendo en cuenta la ecuación. 1. \( 2^{1} a(2 x-1)=3 \frac{9}{2} a \) 2. \( \frac{36+x}{x}=\frac{4 x^{2}}{36+x} \quad \begin{array}{c}\text { usar regla } \\ \text { generaal } \\ x=-b \pm\end{array} \) 3. \( V_{1}^{2} x^{2}+V_{2} x^{2}=d^{2} \) general \( X=\frac{-b \pm}{b^{2}-4 a c} \) \( 2 a \) 4. \( 5^{2}=x^{2}+4^{2} \) 5. \( \frac{30}{26}=\frac{2 x+\frac{1}{2}}{2 x-1} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{30}{26}=\frac{2x+\frac{1}{2}}{2x-1}\) - step1: Find the domain: \(\frac{30}{26}=\frac{2x+\frac{1}{2}}{2x-1},x\neq \frac{1}{2}\) - step2: Reduce the fraction: \(\frac{15}{13}=\frac{4x+1}{2\left(2x-1\right)}\) - step3: Swap the sides: \(\frac{4x+1}{2\left(2x-1\right)}=\frac{15}{13}\) - step4: Cross multiply: \(\left(4x+1\right)\times 13=2\left(2x-1\right)\times 15\) - step5: Simplify the equation: \(13\left(4x+1\right)=30\left(2x-1\right)\) - step6: Calculate: \(52x+13=30\left(2x-1\right)\) - step7: Calculate: \(52x+13=60x-30\) - step8: Move the expression to the left side: \(52x+13-\left(60x-30\right)=0\) - step9: Calculate: \(-8x+43=0\) - step10: Move the constant to the right side: \(-8x=0-43\) - step11: Remove 0: \(-8x=-43\) - step12: Change the signs: \(8x=43\) - step13: Divide both sides: \(\frac{8x}{8}=\frac{43}{8}\) - step14: Divide the numbers: \(x=\frac{43}{8}\) - step15: Check if the solution is in the defined range: \(x=\frac{43}{8},x\neq \frac{1}{2}\) - step16: Find the intersection: \(x=\frac{43}{8}\) Solve the equation \( \frac{36+x}{x}=\frac{4 x^{2}}{36+x} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{36+x}{x}=\frac{4x^{2}}{36+x}\) - step1: Find the domain: \(\frac{36+x}{x}=\frac{4x^{2}}{36+x},x \in \left(-\infty,-36\right)\cup \left(-36,0\right)\cup \left(0,+\infty\right)\) - step2: Cross multiply: \(\left(36+x\right)\left(36+x\right)=x\times 4x^{2}\) - step3: Simplify the equation: \(\left(36+x\right)^{2}=4x^{3}\) - step4: Expand the expression: \(1296+72x+x^{2}=4x^{3}\) - step5: Move the expression to the left side: \(1296+72x+x^{2}-4x^{3}=0\) - step6: Calculate: \(x\approx 7.831349\) - step7: Check if the solution is in the defined range: \(x\approx 7.831349,x \in \left(-\infty,-36\right)\cup \left(-36,0\right)\cup \left(0,+\infty\right)\) - step8: Find the intersection: \(x\approx 7.831349\) Solve the equation \( 2^{1} a(2 x-1)=3 \frac{9}{2} a \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{1}a\left(2x-1\right)=3\frac{9}{2}\times a\) - step1: Evaluate the power: \(2a\left(2x-1\right)=\frac{15}{2}a\) - step2: Divide both sides: \(\frac{2a\left(2x-1\right)}{2a}=\frac{\frac{15}{2}a}{2a}\) - step3: Divide the numbers: \(2x-1=\frac{15}{4}\) - step4: Move the constant to the right side: \(2x=\frac{15}{4}+1\) - step5: Add the numbers: \(2x=\frac{19}{4}\) - step6: Multiply by the reciprocal: \(2x\times \frac{1}{2}=\frac{19}{4}\times \frac{1}{2}\) - step7: Multiply: \(x=\frac{19}{8}\) Solve the equation \( 5^{2}=x^{2}+4^{2} \). Solve the quadratic equation by following steps: - step0: Solve using square roots: \(5^{2}=x^{2}+4^{2}\) - step1: Evaluate the power: \(5^{2}=x^{2}+16\) - step2: Swap the sides: \(x^{2}+16=5^{2}\) - step3: Add or subtract both sides: \(x^{2}=9\) - step4: Simplify the expression: \(x=\pm \sqrt{9}\) - step5: Simplify the expression: \(x=\pm 3\) - step6: Separate into possible cases: \(\begin{align}&x=3\\&x=-3\end{align}\) - step7: Rewrite: \(x_{1}=-3,x_{2}=3\) Solve the equation \( V_{1}^{2} x^{2}+V_{2} x^{2}=d^{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(V_{1}^{2}x^{2}+V_{2}x^{2}=d^{2}\) - step1: Simplify: \(\left(V_{1}^{2}+V_{2}\right)x^{2}=d^{2}\) - step2: Divide both sides: \(\frac{\left(V_{1}^{2}+V_{2}\right)x^{2}}{V_{1}^{2}+V_{2}}=\frac{d^{2}}{V_{1}^{2}+V_{2}}\) - step3: Divide the numbers: \(x^{2}=\frac{d^{2}}{V_{1}^{2}+V_{2}}\) - step4: Simplify the expression: \(x=\pm \sqrt{\frac{d^{2}}{V_{1}^{2}+V_{2}}}\) - step5: Simplify: \(x=\pm \frac{d\sqrt{V_{1}^{2}+V_{2}}}{V_{1}^{2}+V_{2}}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{d\sqrt{V_{1}^{2}+V_{2}}}{V_{1}^{2}+V_{2}}\\&x=-\frac{d\sqrt{V_{1}^{2}+V_{2}}}{V_{1}^{2}+V_{2}}\end{align}\) Aquí están las soluciones para despejar \( x \) en cada una de las ecuaciones proporcionadas: 1. **Ecuación:** \( 2^{1} a(2 x-1)=3 \frac{9}{2} a \) **Despeje:** \[ x = \frac{19}{8} \] 2. **Ecuación:** \( \frac{36+x}{x}=\frac{4 x^{2}}{36+x} \) **Despeje:** \[ x \approx 7.831349 \] 3. **Ecuación:** \( V_{1}^{2} x^{2}+V_{2} x^{2}=d^{2} \) **Despeje:** \[ x = \frac{d\sqrt{V_{1}^{2}+V_{2}}}{V_{1}^{2}+V_{2}} \quad \text{o} \quad x = -\frac{d\sqrt{V_{1}^{2}+V_{2}}}{V_{1}^{2}+V_{2}} \] 4. **Ecuación:** \( 5^{2}=x^{2}+4^{2} \) **Despeje:** \[ x_{1} = -3, \quad x_{2} = 3 \] 5. **Ecuación:** \( \frac{30}{26}=\frac{2 x+\frac{1}{2}}{2 x-1} \) **Despeje:** \[ x = \frac{43}{8} \quad \text{o} \quad x = 5.375 \] Si necesitas más detalles sobre algún paso específico o más ejemplos, no dudes en preguntar.