Example Question - isolating variables
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Solving an Equation for X in Terms of N
Claro, vamos a resolver la ecuación que se muestra en la imagen. La ecuación es 3X - 3N = -3. Para resolver esta ecuación para X en términos de N, queremos despejar X. Esto significa que debemos aislar X en un lado de la ecuación. Aquí están los pasos para hacerlo: 1. Primero, sumamos 3N a ambos lados de la ecuación para mover el término que involucra a N al lado derecho de la ecuación: 3X - 3N + 3N = -3 + 3N Esto se simplifica a: 3X = -3 + 3N 2. Ahora, dividimos ambos lados de la ecuación por 3 para resolver para X: 3X / 3 = (-3 + 3N) / 3 Esto nos da: X = -1 + N Por lo tanto, la expresión simplificada para X en términos de N es X = N - 1.
Solving Linear Equation for y in terms of x
La imagen muestra la ecuación lineal \( 3x - 3y = -3 \). Para resolver esta ecuación para \( y \) en términos de \( x \), sigamos estos pasos: 1. Primero, aislemos el término con \( y \) sumando \( 3y \) a ambos lados de la ecuación para obtener \( 3x = 3y - 3 \). 2. Luego, para despejar \( y \), vamos a pasar \( -3 \) al otro lado del signo igual sumando \( 3 \) a ambos lados, así que ahora tenemos \( 3x + 3 = 3y \). 3. Ahora, para obtener \( y \) por sí solo, dividimos todos los términos de la ecuación por \( 3 \) para obtener \( y \) en función de \( x \): \[ \frac{3x + 3}{3} = \frac{3y}{3} \] Esto nos da: \[ x + 1 = y \] Por lo tanto, \( y = x + 1 \). Esta es la ecuación despejada para \( y \) en términos de \( x \).
Solving a Linear Equation by Order of Operations
Por supuesto, te ayudaré a resolver la ecuación que aparece en la imagen: \( x + 48 \times -20 = 0 \) Primero, debemos resolver la operación de multiplicación antes de proceder con la suma, de acuerdo con el orden de las operaciones (primero multiplicaciones y divisiones, luego sumas y restas). Entonces multiplicamos 48 por -20: \( 48 \times -20 = -960 \) Ahora sustituimos este resultado en la ecuación original: \( x - 960 = 0 \) Para despejar \( x \), necesitamos sumar 960 a ambos lados de la ecuación para aislar la variable \( x \): \( x - 960 + 960 = 0 + 960 \) \( x = 960 \) Por lo tanto, el valor de \( x \) es 960.
Solving Linear Equations
Para resolver la ecuación \( 4x + 17 = 3x + 24 \), necesitamos despejar la variable \( x \). Vamos paso por paso. 1. Primero, queremos tener todos los términos con \( x \) en un lado de la ecuación y los términos sin \( x \) en el otro lado. Para eso, restaremos \( 3x \) de ambos lados de la ecuación para llevar todos los \( x \) al lado izquierdo. \[ 4x + 17 - 3x = 3x + 24 - 3x \] \[ x + 17 = 24 \] 2. Ahora, queremos aislar \( x \), para eso restaremos 17 de ambos lados de la ecuación. \[ x + 17 - 17 = 24 - 17 \] \[ x = 7 \] La solución de la ecuación es \( x = 7 \).
Solving Inequalities with Variable g
The inequality given in the image is: 5g + 3 ≤ 37 To solve for g, we follow these steps: 1. Subtract 3 from both sides to isolate the term with g on one side: 5g + 3 - 3 ≤ 37 - 3 Which simplifies to: 5g ≤ 34 2. Now, divide both sides by 5 to solve for g: \( \frac{5g}{5} \) ≤ \( \frac{34}{5} \) This gives us: g ≤ 6.8 So, any value of g that is less than or equal to 6.8 will satisfy the inequality. Looking at the options provided: - g = 8: this does not satisfy the inequality because 8 is greater than 6.8. - g = 5: this does satisfy the inequality because 5 is less than 6.8. - g = -2: this also satisfies the inequality because -2 is less than 6.8. - g = -1: this too satisfies the inequality because -1 is less than 6.8. Since three of the options satisfy the inequality and we are asked for a solution to the inequality, we have to identify which one of the options is correct. Based on the information provided in the image, it seems we are asked for a single solution, but there are three possible solutions. In this scenario, without further information, it would be best to state that g = 5, g = -2, and g = -1 all are solutions to the given inequality since they all result in a true statement when substituted back into the inequality.
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 Linear Equations
To solve the equation \( y + 6 = 2(x + 9) \), we can follow these steps: 1. Distribute the 2 over the terms inside the parentheses: \( y + 6 = 2 \cdot x + 2 \cdot 9 \) 2. Simplify the equation by performing the multiplication: \( y + 6 = 2x + 18 \) 3. To isolate \(y\), we need to move all terms involving \(y\) to one side of the equation and all terms involving \(x\) to the other side. Since we want to get \(y\) alone, let's subtract 6 from both sides to move all constants to the opposite side of \(y\): \( y + 6 - 6 = 2x + 18 - 6 \) \( y = 2x + 12 \) The equation \( y = 2x + 12 \) is now solved for \(y\) in terms of \(x\).
Solving a Linear Equation
To solve the equation \( y + 6 = 5(x - 7) \), we shall first expand the right side of the equation by distributing the 5 into the parentheses: \[ y + 6 = 5x - 35 \] Next, to solve for \( y \), we want to isolate \( y \) on one side of the equation. We can do this by subtracting 6 from both sides of the equation: \[ y = 5x - 35 - 6 \] \[ y = 5x - 41 \] Now the equation is solved for \( y \) in terms of \( x \). So the final expression is \( y = 5x - 41 \).
Solving Absolute Value Inequality
The inequality provided in the image is \( 8|y| \leq 24 \). To solve for \( y \), we need to isolate \( y \). We start by dividing both sides of the inequality by 8: \( |y| \leq 24 / 8 \\ |y| \leq 3 \) The absolute value sign indicates that whatever value \( y \) takes, when we take its absolute value, it should be less than or equal to 3. This means that \( y \) can be any number in the range of -3 to 3 inclusive. Thus, the solution set for \( y \) is: \( -3 \leq y \leq 3 \)
Solving a Basic Equation
This equation provided in the image is: \[ x + 5 + 6 = 2^3 \] To solve for \( x \), we should first simplify the right side of the equation. \(2^3\) equals \(2 \times 2 \times 2\), which is \(8\). So the equation becomes: \[ x + 5 + 6 = 8 \] Next, combine the constant terms on the left side: \[ x + 11 = 8 \] Now subtract 11 from both sides to isolate \( x \): \[ x = 8 - 11 \] \[ x = -3 \] Therefore, the solution to the equation is \( x = -3 \).
Solving Equations by Addition
To solve the given equations by addition, you'll combine like terms from each equation. Here are the equations: -2x + 4y = 15 -12x - 4y = -8 When you add them together, the terms -2x and -12x combine to make -14x, and the terms 4y and -4y cancel each other out, leaving 0y (which is simply 0). The right side of the equations will be added as well: 15 + (-8). So the sum of the two equations is: -14x + 0 = 15 - 8 Simplify the right side: -14x = 7 Now, to solve for x, you need to isolate x by dividing by the coefficient -14: x = 7 / (-14) Simplifying this fraction: x = -1/2 The value that belongs in the green box is not the value for x but rather the calculated right side of the equation, which is 7. Therefore, the answer to what value belongs in the green box is not directly given by the simplified form of x, but rather by the combined terms on the right side before we divide by -14. So the answer is 7, which corresponds to option B.
Solving a Linear Equation with Multiple Variables
Claro, vamos a resolver la pregunta. Se nos da la ecuación siguiente: \(4T + 5V = X + 3.38 + .8X\) Para simplificar la ecuación, primero combinaremos los términos que contienen "X": \(4T + 5V = 1X + 0.8X + 3.38\) \(4T + 5V = 1.8X + 3.38\) Ahora, queremos despejar "X" en términos de "T" y "V". Para hacer esto, aislamos "X": \(1.8X = 4T + 5V - 3.38\) Dividimos ambos lados de la ecuación por 1.8 para obtener "X": \(X = \frac{4T + 5V - 3.38}{1.8}\) Supongamos que "X" es el monto original que la gerencia pensaba gastar y que el resultado que obtenemos de la ecuación es el monto final después de que la gerente decidió multiplicar el resultado final por 10 veces el valor de la incógnita "V" para los próximos eventos. Entonces, el monto final sería: Monto final \(= 10V \times X\) Monto final \(= 10V \times \frac{4T + 5V - 3.38}{1.8}\) Para resolver nuestro problema completamente, necesitamos conocer los valores específicos de "T" y "V". Sin embargo, dado que no se proporcionan en la solicitud original ni en la imagen, no podemos dar un valor numérico definitivo para el monto final. Si proporcionas los valores de "T" y "V", podré realizar el cálculo final.
Solving Linear Equations for Y and Expressing X as a Function of Y
Tampaknya Anda ingin menyelesaikan persamaan dalam gambar untuk variabel "y." Persamaan yang ditampilkan adalah: \[ y = 3 + 2x \] Ini adalah bentuk persamaan linear di mana "y" sudah diisolasi di satu sisi persamaan. Oleh karena itu, tidak ada "penyelesaian" lebih lanjut yang diperlukan untuk "y" karena sudah dalam bentuk yang dipecahkan. Namun, jika Anda ingin mengekspresikannya dalam bentuk "x," kita hanya perlu mengekspresikan "x" sebagai fungsi dari "y." Jika kita ingin mengekspresikan "x" dalam persamaan tersebut, langkahnya adalah sebagai berikut: \[ y = 3 + 2x \] Kemudian, kita dapat mengurangi kedua sisi dengan 3 untuk mendapatkan: \[ y - 3 = 2x \] Dan terakhir, kita bagi kedua sisi dengan 2 untuk menyelesaikan "x": \[ \frac{y - 3}{2} = x \] Jadi, "x" dinyatakan sebagai \[ x = \frac{y - 3}{2} \] dalam bentuk "y."
Solving Linear Equations for y
The image shows a step-by-step process of solving a linear equation for y. The initial equation given is: 3x + 2y = 10 The first step is to isolate y by moving the term involving x to the other side of the equation. This is done by subtracting 3x from both sides of the equation: 2y = 10 - 3x Then, to solve for y, divide both sides of the equation by 2: y = (10 - 3x) / 2 This results in the final expression for y: y = 5 - (3/2)x This equation now expresses y in terms of x.
Solving Linear Equations
The given equation is \( -3x - 3y = 3 \). To solve for one variable in terms of the other, you can isolate one variable on one side. Let's solve for \( y \) in terms of \( x \): First, add \( 3x \) to both sides to get: \[ -3y = 3x + 3 \] Next, divide each term by \( -3 \) to solve for \( y \): \[ y = -x - 1 \] Now, \( y \) is expressed in terms of \( x \). If you want to solve for \( x \) in terms of \( y \) instead, you would do a similar process but isolate \( x \) instead of \( y \).
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