Example Question - algebraic equation
Here are examples of questions we've helped users solve.
Finding the Value of a Variable
<p>Primero, expandimos cada término en la ecuación:</p> <p>3(x + 2) + 2(x - 3) = 6(x - 5)</p> <p>Esto se convierte en:</p> <p>3x + 6 + 2x - 6 = 6x - 30</p> <p>Combinando términos similares:</p> <p>5x = 6x - 30</p> <p>Ahora, restamos 6x de ambos lados:</p> <p>5x - 6x = -30</p> <p>-x = -30</p> <p>Multiplicamos por -1:</p> <p>x = 30</p>
Finding the Original Number Before Subtraction and Multiplication
Seja \( x \) o número original. De acordo com o enunciado, se você subtrai 8 do número e depois multiplica por 2, o produto é 14. Montamos a seguinte equação para representar a situação: <p>\( (x - 8) \cdot 2 = 14 \)</p> Dividimos ambos os lados da equação por 2: <p>\( x - 8 = \frac{14}{2} \)</p> <p>\( x - 8 = 7 \)</p> Agora, adicionamos 8 a ambos os lados da equação para isolar \( x \): <p>\( x = 7 + 8 \)</p> <p>\( x = 15 \)</p> Portanto, o número original é 15.
Calculating Specific Expansion Terms and Investigating Geometric Mean Relation
For the 2nd question (problem 8), since it is cut off, we are unable to provide a complete solution. However, for the 3rd question (problem 9), we can proceed: <p>The sum of the two numbers is $6$ times their geometric mean, then </p> <p>Let the two numbers be $a$ and $b$. Given that $a+b = 6\sqrt{ab}$ and we are to show that $a$ and $b$ are in the ratio $(3 + 2\sqrt{2}):(3 - 2\sqrt{2})$.</p> <p>Since we have a ratio of sums to a geometric mean, let's assume $a = (3 + 2\sqrt{2})k$ and $b = (3 - 2\sqrt{2})k$, where $k$ is some positive constant.</p> <p>\[ a + b = (3 + 2\sqrt{2})k + (3 - 2\sqrt{2})k\]</p> <p>\[ a + b = 3k + 2\sqrt{2}k + 3k - 2\sqrt{2}k\]</p> <p>\[ a + b = 6k\]</p> <p>The geometric mean of $a$ and $b$ is $\sqrt{ab}$:</p> <p>\[\sqrt{ab} = \sqrt{(3 + 2\sqrt{2})k \cdot (3 - 2\sqrt{2})k}\]</p> <p>\[\sqrt{ab} = \sqrt{(9 - 8)k^2}\]</p> <p>\[\sqrt{ab} = k\]</p> <p>Now, we compare the sum and geometric mean:</p> <p>\[6k = 6\sqrt{ab}\]</p> <p>\[k = \sqrt{ab}\]</p> <p>This confirms that the sum of $a$ and $b$ is indeed $6$ times their geometric mean, and thus the numbers are in the desired ratio.</p> For question 9, which discusses mean and standard deviation, no specific mathematical equations are shown, and thus a solution cannot be provided based on the information given in the image.
Solving a Linear Equation with Fraction
<p>\(\frac{2x + 5}{3} = 11\)</p> <p>Multiply both sides by 3 to eliminate the fraction:</p> <p>\(2x + 5 = 33\)</p> <p>Subtract 5 from both sides:</p> <p>\(2x = 28\)</p> <p>Divide both sides by 2:</p> <p>\(x = 14\)</p>
Analysis of Different Approaches to Solving an Algebraic Equation
<p>Para María, su procedimiento es como sigue:</p> <p>Expandir y simplificar la ecuación dada:</p> \[ (x + 2)(x + 3) = 5(x + 3) \] \[ x^2 + 3x + 2x + 6 = 5x + 15 \] \[ x^2 + 5x + 6 = 5x + 15 \] <p>Restar \(5x + 15\) de ambos lados:</p> \[ x^2 + 5x + 6 - (5x + 15) = 0 \] \[ x^2 + 6 - 15 = 0 \] \[ x^2 - 9 = 0 \] <p>Factorizar la diferencia de cuadrados:</p> \[ (x + 3)(x - 3) = 0 \] <p>Solucionar cada factor igualado a cero:</p> \[ x + 3 = 0 \quad \text{or} \quad x - 3 = 0 \] \[ x = -3, \quad x = 3 \]</p> <p>María encuentra correctamente las soluciones \( x = -3 \) y \( x = 3 \).</p> <p>Para Nelson, su procedimiento es como sigue:</p> <p>Expandir la ecuación dada:</p> \[ (x + 2)(x + 3) - 5(x + 3) = 0 \] \[ x^2 + 3x + 2x + 6 - 5x - 15 = 0 \] \[ x^2 + 6 - 15 = 0 \] <p>Esta simplificación es incorrecta, ya que se ha omitido el término \( x \) presente en la expansión:</p> \[ x^2 + 5x - 9 = 0 \] <p>El error de Nelson es que no simplificó correctamente los términos \( x \).</p> <p>Para Oscar, su procedimiento es como sigue:</p> <p>Dividir ambos lados de la ecuación original por \( x + 3 \), suponiendo que \( x + 3 \neq 0 \):</p> \[ \frac{(x + 2)(x + 3)}{x + 3} = \frac{5(x + 3)}{x + 3} \] \[ x + 2 = 5 \] <p>Restar 2 de ambos lados:</p> \[ x = 3 \] <p>Oscar encuentra la solución \( x = 3 \), pero al dividir por \( x + 3 \), omitió la solución \( x = -3 \), cuando \( x + 3 = 0 \).</p> <p>Por lo tanto, la solución completa de la ecuación es \( x = -3, x = 3 \), y la respuesta correcta es la proporcionada por María.</p>
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 an Algebraic Equation Step by Step
Por supuesto, vamos a resolver la ecuación paso a paso. La ecuación dada es: \[ 2(1 + 2x) = 10 \] Primero, distribuimos el 2 que está fuera del paréntesis: \[ 2 \cdot 1 + 2 \cdot 2x = 10 \] \[ 2 + 4x = 10 \] Ahora, vamos a aislar la variable \( x \): Primero, restamos 2 de ambos lados de la ecuación para deshacernos del número que está añadido a \( 4x \): \[ 4x = 10 - 2 \] \[ 4x = 8 \] Finalmente, dividimos ambos lados de la ecuación por 4 para encontrar el valor de \( x \): \[ x = \frac{8}{4} \] \[ x = 2 \] Así que la solución de la ecuación es \( x = 2 \).
Solving Linear Equation for x
The equation in the image is: \[x + 3 = 5x - 1\] To solve the equation for \(x\), follow these steps: 1. Get all terms containing \(x\) on one side of the equation and the constant terms on the other side. You can do this by subtracting \(x\) from both sides to get all the \(x\)'s on one side, and adding \(1\) to both sides to get all the constants on the other side. \[x - x + 3 + 1 = 5x - x - 1 + 1\] Simplifying this gives: \[4 = 4x\] 2. Now, to solve for \(x\), divide both sides of the equation by \(4\): \[\frac{4}{4} = \frac{4x}{4}\] This simplifies to: \[1 = x\] So, the solution to the equation is: \[x = 1\]
Solving an Algebraic Equation with a Contradiction
The image shows the following algebraic equation: -4x + 8x - 3 = 8x - 1/2 (8x - 4) Let's solve this step by step. Firstly, combine like terms on the left side of the equation: 4x - 3 = 8x - 1/2 (8x - 4) Now we need to distribute the 1/2 on the right side of the equation: 4x - 3 = 8x - (1/2 * 8x) + (1/2 * 4) Simplify the products: 4x - 3 = 8x - 4x + 2 Combine like terms on the right side of the equation: 4x - 3 = 4x + 2 Now, we can subtract 4x from both sides to isolate the constant terms: 4x - 4x - 3 = 4x - 4x + 2 Which simplifies to: -3 = 2 This results in a clear contradiction as -3 cannot equal 2. Therefore, there is no solution to the equation, indicating that it is inconsistent. This type of equation is also known as a contradiction because it indicates that there was probably a mistake in the formulation of the original problem or it's a system with no intersection points.
Solving an Algebraic Equation for Variable
It appears that you would like to solve the algebraic equation: \[ \frac{2ac + 14}{2a} = 9 \] To solve for c, follow these steps: 1. Multiply both sides of the equation by \(2a\) to get rid of the denominator: \[ 2ac + 14 = 18a \] 2. Subtract 14 from both sides to isolate terms with \(c\) on one side: \[ 2ac = 18a - 14 \] 3. Divide both sides by \(2a\) to solve for \(c\): \[ c = \frac{18a - 14}{2a} \] 4. Simplify the right side if possible: \[ c = 9 - \frac{14}{2a} \] \[ c = 9 - \frac{7}{a} \] So, \(c = 9 - \frac{7}{a}\), assuming \(a \neq 0\).
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