Example Question - variable isolation
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Algebraic Equation Simplification
<p>La ecuación proporcionada en la imagen es: \(5x - (-8) + (-9) = 9x - (-7 + 1)\)</p> <p>Primero simplificamos la ecuación combinando términos semejantes y eliminando los paréntesis:</p> <p>\(5x + 8 - 9 = 9x - (-7 + 1)\)</p> <p>\(5x - 1 = 9x - (6)\)</p> <p>Ahora procedemos a aislar la variable \(x\):</p> <p>\(5x - 9x = -6 + 1\)</p> <p>\(-4x = -5\)</p> <p>\(x = \frac{-5}{-4}\)</p> <p>\(x = \frac{5}{4}\)</p> <p>Por lo tanto, la solución de la ecuación es \(x = \frac{5}{4}\).</p>
Solving a Linear Equation with Variables on Both Sides
<p>Begin by expanding the terms within the parentheses:</p> <p>\[2(3x + 5) - 3(3x - 1) = 3(4 - x)\]</p> <p>\[6x + 10 - 9x + 3 = 12 - 3x\]</p> <p>Combine like terms:</p> <p>\[-3x + 13 = 12 - 3x\]</p> <p>Add \(3x\) to both sides to move variables to one side:</p> <p>\[-3x + 3x + 13 = 12 - 3x + 3x\]</p> <p>\[13 = 12\]</p> <p>Since 13 is not equal to 12, we have an inconsistency. The equation has no solution because it is not an identity, and the two sides of the equation are not equal for any value of \(x\).</p>
Solving a Basic Algebraic Equation for a Variable
<p>To solve the equation \( y(x - 1) = z \) for \( x \), follow these steps:</p> <p>1. Distribute \( y \) into the parentheses: \( yx - y = z \)</p> <p>2. Add \( y \) to both sides of the equation: \( yx = z + y \)</p> <p>3. Divide both sides by \( y \) to isolate \( x \): \( x = \frac{z + y}{y} \)</p>
Solving for x in an Algebraic Equation
<p>If \(y(x-1)=z\) then to solve for \(x\), we need to isolate \(x\). We begin by dividing both sides of the equation by \(y\) to get:</p> <p>\(x-1 = \frac{z}{y}\)</p> <p>Next, we add 1 to both sides of the equation to solve for \(x\):</p> <p>\(x = \frac{z}{y} + 1\)</p>
Solving Equation by Despelling Y
Para resolver la ecuación que se muestra en la imagen, primero debemos decidir qué variable queremos despejar. Podemos despejar X o Y. Voy a mostrarte cómo despejar Y, pero el proceso sería similar si despejáramos X. La ecuación es: 6X + 2Y = 4 Para despejar Y, seguimos estos pasos: 1. Restamos 6X de ambos lados de la ecuación para aislar los términos con Y en un lado: 2Y = 4 - 6X 2. Dividimos ambos lados de la ecuación por 2 para obtener Y solo: Y = (4 - 6X) / 2 3. Simplificamos la expresión dividiendo ambos términos del numerador por 2: Y = 2 - 3X Entonces, hemos despejado Y en términos de X. La fórmula resultante Y = 2 - 3X nos da el valor de Y para cualquier valor de X que elijamos.
Solving Inequalities with Variable Isolation
To solve the inequality \(5g + 3 \leq 37\), we need to isolate the variable \(g\). Here are the steps: 1. Subtract 3 from both sides: \[5g + 3 - 3 \leq 37 - 3\] \[5g \leq 34\] 2. Divide both sides by 5: \[\frac{5g}{5} \leq \frac{34}{5}\] \[g \leq \frac{34}{5}\] \[g \leq 6.8\] Now, let's look at the answer choices to find which ones are less than or equal to 6.8: - \(g = -8\) is less than 6.8; hence it's a valid solution. - \(g = -5\) is less than 6.8; hence it's a valid solution. - \(g = -2\) is less than 6.8; hence it's a valid solution. - \(g = -1\) is less than 6.8; hence it's a valid solution. All of the given choices \(g = -8, g = -5, g = -2, g = -1\) are solutions to the inequality \(5g + 3 \leq 37\).
Solving Linear Inequalities by Isolating Variables
To solve the inequality 5q + 3 < 37, you need to isolate the variable q. Here are the steps: 1. Subtract 3 from both sides: 5q + 3 - 3 < 37 - 3 2. This simplifies to: 5q < 34 3. Now, divide both sides by 5: 5q / 5 < 34 / 5 4. You get: q < 6.8 Now, let's look at the provided options: - \( q = -8 \) is less than 6.8, so it is a solution. - \( q = -5 \) is also less than 6.8, so it is a solution. - \( q = -2 \) is less than 6.8, so it is a solution. - \( q = -1 \) is less than 6.8, so it is also a solution. All four options are solutions to the inequality because they are all less than 6.8. However, if the question intends to ask for only one solution, they typically expect the greatest value that satisfies the inequality. Since \( q = -1 \) is the greatest value among the choices and still satisfies the inequality \( q < 6.8 \), it would be considered the best solution from the provided options.
Solving Inequality 5q + 3 < 37
To solve the inequality 5q + 3 < 37, follow these steps: 1. Subtract 3 from both sides to isolate the term with the variable q: 5q + 3 - 3 < 37 - 3 5q < 34 2. Divide both sides by 5 to solve for q: 5q/5 < 34/5 q < 6.8 Now let's check the provided options to determine which one is a solution: - q = -8: This is indeed less than 6.8, so it is a possible solution. - q = -5: This is also less than 6.8, so it is a possible solution. - q = -2: Similarly, this is less than 6.8, so it is a possible solution. - q = -1: This, too, is less than 6.8, so it is a possible solution. All the provided options are less than 6.8. Therefore, all of the given options are solutions to the inequality 5q + 3 < 37.
Step-by-Step Equation Solving
Por supuesto, vamos a resolver la ecuación paso a paso. La ecuación que nos has dado es: \[ 2x - 1 = 5x + 8 \] Para resolver la ecuación, primero queremos despejar la variable \( x \) en uno de los lados de la ecuación. Hagámoslo restando \( 2x \) en ambos lados y también restando \( 8 \) en ambos lados para aislar \( x \). Así que tenemos: Restamos \( 2x \) de ambos lados y obtenemos: \[ -1 = 3x + 8 \] Ahora, restamos \( 8 \) de ambos lados y obtenemos: \[ -1 - 8 = 3x \] \[ -9 = 3x \] Finalmente, dividimos ambos lados entre 3 para despejar \( x \): \[ \frac{-9}{3} = x \] \[ x = -3 \] La solución a la ecuación es \( x = -3 \).
Solving an Equation Step by Step
Claro, vamos a resolver la ecuación paso a paso. Tienes la ecuación \(2x - 1 = 6x + 8\). El primer paso es llevar todas las variables a un lado de la ecuación y todos los términos constantes al otro lado. Para hacer esto, podemos restar \(2x\) de ambos lados de la ecuación y sustraer \(8\) de ambos lados al mismo tiempo. Así hacemos ambas operaciones en un paso y nos queda: \(2x - 2x - 1 - 8 = 6x - 2x + 8 - 8\) Esto se simplifica a: \(-1 - 8 = 4x\) Sumamos -1 y -8: \(-9 = 4x\) Para despejar \(x\), dividimos ambos lados de la ecuación entre 4: \(\frac{-9}{4} = \frac{4x}{4}\) Lo que nos deja con: \(x = \frac{-9}{4}\) Por lo tanto, la solución de la ecuación es \(x = -\frac{9}{4}\) o lo que es lo mismo, \(x = -2.25\) si prefieres la respuesta en forma decimal.
Solving a Linear Equation
Para resolver la ecuación, primero simplifiquemos la expresión dentro de los paréntesis: \( -5y(3 + 5) = 10 \) Multiplicamos los números dentro de los paréntesis: \( -5y(8) = 10 \) Ahora, resolvemos la multiplicación: \( -40y = 10 \) Para aislar la variable \( y \), dividimos ambos lados de la ecuación por \( -40 \): \( y = \frac{10}{-40} \) Simplificamos la fracción dividiendo el numerador y el denominador por \( 10 \): \( y = \frac{1}{-4} \) Por lo tanto, la solución de la ecuación es: \( y = -\frac{1}{4} \) o \( y = -0.25 \).
Solving a Simple Equation
To solve the equation given in the image, which is \( x + 5 + 6 = 2^3 \), you need to simplify and solve for x. First, simplify the right side of the equation: \( 2^3 = 2 \times 2 \times 2 = 8 \) So the equation becomes: \( x + 5 + 6 = 8 \) Next, combine like terms on the left side: \( x + 11 = 8 \) Now, to solve for x, you need to isolate the variable by subtracting 11 from both sides of the equation: \( x + 11 - 11 = 8 - 11 \) \( x = -3 \) Hence, the solution to the equation is \( x = -3 \).
Solving an Equation with One Variable
Để tìm giá trị của x, bạn sẽ cần giải phương trình \( x \times 5,6 = 19,04 \). Bước 1: Phân chia cả hai phía của phương trình cho 5,6 để tách x ra một mình. \[ x = \frac{19,04}{5,6} \] Bước 2: Tính toán phép chia. \[ x = \frac{19,04}{5,6} = 3,4 \] Vậy giá trị của x là 3,4.
Solving an Equation with Fractional Coefficients
The equation given in the image is \( 3 + \frac{3}{2}k = 4k - 2 \). To solve for \( k \), follow these steps: 1. First, try to get all the terms with \( k \) on one side of the equation and the constant terms on the other side. 2. Subtract \( \frac{3}{2}k \) from both sides to move the \( k \) terms to the right side: \( 3 = 4k - \frac{3}{2}k - 2 \) 3. Combine like terms on the right side. To combine \( 4k \) and \( -\frac{3}{2}k \), you can convert \( 4k \) to \( \frac{8}{2}k \) to have a common denominator: \( 3 = \left( \frac{8}{2}k - \frac{3}{2}k \right) - 2 \) 4. Now, subtract the coefficients: \( 3 = \frac{5}{2}k - 2 \) 5. Next, add 2 to both sides to move the constant terms to the left side: \( 3 + 2 = \frac{5}{2}k \) 6. Combine the constants: \( 5 = \frac{5}{2}k \) 7. Now, to solve for \( k \), multiply both sides of the equation by the reciprocal of \( \frac{5}{2} \), which is \( \frac{2}{5} \): \( 5 \times \frac{2}{5} = \frac{5}{2}k \times \frac{2}{5} \) 8. This simplifies nicely, as the \( \frac{5}{2} \) and \( \frac{2}{5} \) cancel each other on the right side: \( k = 5 \times \frac{2}{5} \) 9. After cancellation, we get: \( k = 2 \) So, the solution to the equation is \( k = 2 \).
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