Example Question - fractional equation
Here are examples of questions we've helped users solve.
Solving a Fractional Equation Step by Step
Por supuesto, comenzaré resolviendo la ecuación paso a paso. La ecuación inicial es: \[ \frac{x}{2} + \frac{x}{5} = 2 + \frac{x}{3} - \frac{3}{5} \] Para resolverla, es útil encontrar un denominador común para todos los términos que contienen fracciones. El denominador común más pequeño que podemos usar aquí es 30, ya que 2, 3 y 5 son todos factores de 30. Multiplicamos ambos lados de la ecuación por 30 para eliminar los denominadores: \[ 30 \left( \frac{x}{2} \right) + 30 \left( \frac{x}{5} \right) = 30 \cdot 2 + 30 \left( \frac{x}{3} \right) - 30 \left( \frac{3}{5} \right) \] Esto nos da: \[ 15x + 6x = 60 + 10x - 18 \] Combinamos términos semejantes: \[ 21x = 10x + 42 \] Ahora, restamos 10x de ambos lados para tener los términos con x en un solo lado: \[ 11x = 42 \] Finalmente, dividimos ambos lados por 11 para despejar x: \[ x = \frac{42}{11} \] Entonces, la solución de la ecuación es: \[ x = \frac{42}{11} \approx 3.818 \] Así que x es aproximadamente 3.818.
Solving a Fractional Equation with Variables
The given expression seems to be a mathematical fraction which is missing the operator (like addition, subtraction, multiplication, or division) between the two terms in the numerator. If we assume that the desired operation is addition (which is common), then the expression reads: \[ \frac{1}{x} + \frac{1}{y} = 1 \] To solve this equation, you would normally need additional information about either x or y, since there are infinite solutions to this equation without additional constraints. However, we can manipulate the equation to express y in terms of x or vice versa. Let's solve for y in terms of x: \[ 1 = \frac{1}{x} + \frac{1}{y} \] Subtract \( \frac{1}{x} \) from both sides to isolate \( \frac{1}{y} \): \[ 1 - \frac{1}{x} = \frac{1}{y} \] Now take the reciprocal of both sides to solve for y: \[ y = \frac{1}{1 - \frac{1}{x}} \] To simplify further, you'd find a common denominator in the numerator: \[ y = \frac{1}{\frac{x - 1}{x}} \] Now you can see that y is the reciprocal of \( \frac{x - 1}{x} \), so we flip the fraction: \[ y = \frac{x}{x - 1} \] This gives us y in terms of x. Again, without specific values or additional equations, we cannot find numerical solutions for x and y. But this manipulation gives us a relationship between the two variables.
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