Example Question - equation with fractions
Here are examples of questions we've helped users solve.
Solving an Equation Involving Fractions
Here are the steps to solve the given equation: 1. Simplify the left side of the equation: \( 4 \frac{5}{x+y} = 4 + \frac{20}{x+y} \) 2. Equate the left side to the right side: \( 4 + \frac{20}{x+y} = \frac{25x+y}{x+y} \) 3. Multiply both sides by \( x+y \) to clear the fraction: \( (4 + \frac{20}{x+y})(x+y) = (25x+y) \) 4. Expand and simplify: \( 4(x+y) + 20 = 25x+y \) 5. Distribute the 4: \( 4x + 4y + 20 = 25x+y \) 6. Move terms of x and y to one side: \( 4x - 25x + 4y - y = -20 \) 7. Combine like terms: \( -21x + 3y = -20 \) 8. Divide by -21: \( x = \frac{3y + 20}{21} \) The answer is \( x = \frac{3y + 20}{21} \).
Solving a Mathematical Equation with Fractions
La imagen muestra un problema matemático que requiere resolver una ecuación para encontrar el valor de la variable \( x \). La ecuación es: \( \frac{x-2}{5} - \frac{2x+5}{8} = \frac{x}{9} \) Para resolver esta ecuación, primero se debe encontrar un denominador común para combinar los términos fraccionarios. En este caso, el mínimo común múltiplo (mcm) de los denominadores \(5\), \(8\) y \(9\) es \(360\). Por lo tanto, multiplicamos cada término por \(360\) para eliminar los denominadores: \( \left(\frac{x-2}{5}\right)360 - \left(\frac{2x+5}{8}\right)360 = \left(\frac{x}{9}\right)360 \) Simplificamos cada término: \( 72 \times (x-2) - 45 \times (2x+5) = 40 \times x \) Ahora distribuimos cada coeficiente dentro de los paréntesis: \( 72x - 144 - 90x - 225 = 40x \) Combinamos términos semejantes y llevamos los términos que contienen a \( x \) al mismo lado de la ecuación: \( 72x - 90x - 40x = 144 + 225 \) \( -58x = 369 \) Ahora dividimos ambos lados de la ecuación por \( -58 \) para aislar \( x \): \( x = \frac{369}{-58} \) Finalmente, simplificamos la fracción dividiendo ambos, el numerador y el denominador, por 1 para mantener la misma razón, llegando al resultado: \( x = -6.36 \) Por lo tanto, el valor de \( x \) que satisface la ecuación original es aproximadamente \( -6.36 \).
Solving a Linear Equation with Fractions
The equation in the image is: \( \frac{1}{2}(3x + 6) - \frac{1}{3}(2x - 4) = 20 \) To solve this equation for x, follow these steps: 1. Distribute the fractions to the terms inside the parentheses: \( \frac{1}{2} \cdot 3x + \frac{1}{2} \cdot 6 - \frac{1}{3} \cdot 2x + \frac{1}{3} \cdot 4 = 20 \) 2. Simplify the distributed terms: \( \frac{3}{2}x + 3 - \frac{2}{3}x + \frac{4}{3} = 20 \) 3. Combine like terms: \( \frac{3}{2}x - \frac{2}{3}x = 20 - 3 - \frac{4}{3} \) 4. To combine the x terms, find a common denominator, which in this case is 6: \( \frac{9}{6}x - \frac{4}{6}x = 17 - \frac{4}{3} \) 5. Simplify the x terms and convert 17 into a fraction with the same denominator as 4/3 to continue simplifying: \( \frac{5}{6}x = \frac{51}{3} - \frac{4}{3} \) 6. Simplify the right side of the equation: \( \frac{5}{6}x = \frac{47}{3} \) 7. To solve for x, multiply by the reciprocal of 5/6, which is 6/5: \( x = \frac{47}{3} \cdot \frac{6}{5} \) 8. Multiply the two fractions: \( x = \frac{47 \cdot 6}{3 \cdot 5} \) 9. Simplify the multiplication: \( x = \frac{282}{15} \) 10. Finally, simplify the fraction if possible: \( x = \frac{47 \cdot 2}{3 \cdot 5} \) \( x = \frac{94}{15} \) \( x = 6 \frac{4}{15} \) So, the solution to the equation is \( x = 6 \frac{4}{15} \).
Solving a Cubic Equation with Fractions
To solve the equation \( x + \frac{1}{3x} = \frac{1}{x} - \frac{1}{3} \), you can follow these steps: 1. First, find a common denominator to combine the fractions on each side of the equation. The common denominator for the terms involving \(x\) would be \(3x^2\). 2. Multiply each term on both sides by \(3x^2\) to clear the fractions. This gives us: \[ 3x^3 + 1 = 3x - x^2 \] 3. Rearrange the equation to form a cubic equation by moving all terms to one side: \[ 3x^3 + x^2 - 3x + 1 = 0 \] 4. Now solve this cubic equation for \(x\). Solving cubic equations analytically can be complex, involving Cardano's formula or other methods, but you can also solve it numerically or by factoring if possible. If you want to solve it by factoring, you'd look for values of \(x\) that make the equation equal to zero. This can be done by trial and error (plugging in small integers for \(x\) to see if they work). However, for this particular equation, factoring by inspection may be difficult, and it might be more efficient to use numerical methods or a graphing calculator to find the roots of the equation. If you find that the equation has a rational root, you could use the Rational Root Theorem to help find factors. The Rational Root Theorem states that any rational solution, when written in lowest terms \(p/q\), must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient. In this case, \(p\) would be a factor of 1, and \(q\) would be a factor of 3. Possible rational roots could be \(\pm1, \pm\frac{1}{3}\). But for a precise solution without a calculator, you would need to apply complex methods beyond the scope of this explanation. If the equation were to have a simpler form, or if it were given that a certain method should be used, the approach might differ.
Solving for x in an Equation Involving Fractions
The equation is: \[ \frac{2(3x - 4y)}{x} = 10 \] To solve for \( x \), follow these steps: 1. Multiply both sides by \( x \) to remove the denominator: \[ 2(3x - 4y) = 10x \] 2. Expand the left side: \[ 6x - 8y = 10x \] 3. Rearrange the equation to get all terms involving \( x \) on one side: \[ 6x - 10x = 8y \] 4. Combine like terms: \[ -4x = 8y \] 5. Divide both sides by -4 to solve for \( x \): \[ x = \frac{8y}{-4} \] \[ x = -2y \] Therefore, \( x \) equals negative two times \( y \).
Solving Quadratic Equation with Fractions
To solve the equation \( x + 2 - \frac{15}{x} = 0 \), we can start by finding a common denominator to combine the terms on the left-hand side of the equation. Since "x" is in the denominator of one of the terms, we will use "x" as the common denominator. Here are the steps: 1. Multiply each term by "x" to eliminate the fraction. \[ x(x) + 2(x) - \frac{15(x)}{x} = 0(x) \] 2. Simplify the equation. \[ x^2 + 2x - 15 = 0 \] Now we have a quadratic equation in standard form, which we can solve either by factoring, completing the square, or using the quadratic formula. Let's try factoring first: 3. Factor the quadratic. \[ (x + 5)(x - 3) = 0 \] 4. Use the Zero Product Property which states that if a product of factors is zero, then at least one of the factors must be zero. \[ x + 5 = 0 \quad \text{or} \quad x - 3 = 0 \] 5. Solve for "x". \[ x = -5 \quad \text{or} \quad x = 3 \] Therefore, the solutions to the equation are \( x = -5 \) and \( x = 3 \).
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