Example Question - step by step solution
Here are examples of questions we've helped users solve.
Solving an Inconsistent Equation
Claro, voy a resolver la ecuación paso por paso. La ecuación dada es: 5 + 2(1 - x) = -2x + 3 Comenzaremos distribuyendo el 2 que está multiplicando al paréntesis en la parte izquierda de la ecuación: 5 + 2*1 - 2*x = -2x + 3 5 + 2 - 2x = -2x + 3 Ahora simplificamos sumando 5 y 2 en el lado izquierdo: 7 - 2x = -2x + 3 Después queremos deshacernos de los términos con x en uno de los lados. Sin embargo, observamos que el coeficiente de x es igual en ambos lados de la ecuación (-2x), por lo que al intentar eliminar x, estos términos se cancelarían mutuamente. Prosigamos y veamos qué sucede: Si sumamos 2x a ambos lados para intentar deshacernos de los x, obtenemos: 7 - 2x + 2x = -2x + 2x + 3 7 = 3 Esto nos lleva a una contradicción porque sabemos que 7 no es igual a 3. Esto indica que la ecuación original no tiene solución. En otras palabras, no hay un valor de x que satisfaga la ecuación inicial. Entonces, la ecuación no tiene solución, lo que significa que es una ecuación inconsistente.
Step-by-Step Arithmetic Operation Solution
Por supuesto, vamos a resolver la operación paso a paso: Primero, realizamos las operaciones dentro de los paréntesis. En el primer paréntesis, tenemos una división y una multiplicación: \(-5 \cdot (4 \div -2)\) Vamos a dividir 4 entre -2, lo que nos da: \(-5 \cdot (-2)\) Ahora multiplicamos -5 por -2, como la regla de los signos indica que menos por menos es más, el resultado es: \(10\) Luego, resolvemos la operación dentro del segundo paréntesis: \(-6 - 8\) Lo que nos da como resultado: \(-14\) Ahora, la operación que nos queda es: \(10 - (-14)\) Al restar un número negativo, es como sumar su positivo, por lo que la operación se convierte en: \(10 + 14\) Finalmente sumamos: \(10 + 14 = 24\) Así que el resultado final de la operación es: \(24\)
Complex Mixture Ratio Problem Solving
To solve this problem, we need to work step by step, adjusting the mixture according to the changes described, and using the variable \(x\) to represent the initial volume of the mixture. Let’s break the problem down: 1. Initial Mixture Ratio (water to cordial): 7:2. This means that for every 7 parts water, there are 2 parts cordial. 2. Adding Cordial to Change Ratio to 4:5. In the initial mixture ratio, the total parts are \(7+2=9\). If the original volume of the mixture is \(x\) ml, then we have \(x \cdot \frac{7}{9}\) ml of water and \(x \cdot \frac{2}{9}\) ml of cordial. To make the new mixture have a ratio of 4:5, we assume that the amount of water remains constant, while the amount of cordial is increased to make it equal to \(\frac{5}{4}\) times the amount of water. 3. Sam's Mum Takes a 100 ml Sip. Sam's mum takes a sip from the mixture after cordial has been added but before any more water has been added. So, the ratio remains the same, but the volume decreases by 100 ml. 4. Diluting the Mixture to a Ratio of 3:1 with Water. After Sam's mum takes the 100 ml sip, the mixture is diluted with water to make a new ratio of 3:1. Now let's perform the calculations step by step: Initial mixture (step 1): - Water: \(x \cdot \frac{7}{9}\) ml - Cordial: \(x \cdot \frac{2}{9}\) ml Additional cordial needed to get 4:5 ratio (step 2): We want the ratio of water to cordial to be 4:5, where the water remains the same. We can set up a proportion to find the new amount of cordial (let's call it \(C\)): \[\frac{4}{5} = \frac{x \cdot \frac{7}{9}}{C}\] Solving for \(C\): \[C = x \cdot \frac{7}{9} \cdot \frac{5}{4}\] \[C = x \cdot \frac{35}{36}\] The added cordial amount will be the difference between the new and the old cordial amount: \[C_{\text{added}} = C - x \cdot \frac{2}{9} = x \cdot \frac{35}{36} - x \cdot \frac{2}{9}\] After the sip from Sam's mum (step 3), the volume is reduced by 100 ml but the ratio remains 4:5. Final dilution to 3:1 ratio (step 4): Now, the total volume (\(V_F\)) after dilution will be: \[V_F = \text{current volume} + \text{added water}\] The current volume after the sip is: \[V_{\text{current}} = x + C_{\text{added}} - 100\] To get the exact amount of added water, we need to find the volume of cordial after Sam's mum takes her sip, which represents \(\frac{5}{9}\) of the total mixture then. Let's call this volume \(C_{\text{current}}\). We solve for \(C_{\text{current}}\) using the ratio after the sip: \[C_{\text{current}} = \frac{5}{9} \cdot (V_{\text{current}})\] Now the final volume \(V_F\) should be such that water to cordial ratio is 3:1. Let the added water be \(W_{\text{added}}\), so: \[W_{\text{added}} + x \cdot \frac{7}{9} = 3 \cdot C_{\text{current}}\] Where \(C_{\text{current}}\) is the amount of cordial after Sam's mum's sip, which is still \(x \cdot \frac{35}{36} - 100 \cdot \frac{5}{9}\), since the sip removed 100 ml of the mixture which had 4 parts of water and 5 parts cordial (out of 9 parts total). Combining the equations, you can solve for \(W_{\text{added}}\) and \(V_F\). \[V_F = x + (x \cdot \frac{35}{36} - x \cdot \frac{2}{9}) - 100 + W_{\text{added}}\] Substitute \(W_{\text{added}}\) from the water to cordial ratio equation into the above to get the final volume \(V_F\) in terms of \(x\). It's quite a complex problem with several steps, and each step must be handled carefully to ensure the ratios and volumes add up correctly.
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