CamTutor

Example Question - equation simplification

Here are examples of questions we've helped users solve.

Algebraic Equation Simplification

La ecuación proporcionada en la imagen es: \(5x - (-8) + (-9) = 9x - (-7 + 1)\) Primero simplificamos la ecuación combinando términos semejantes y eliminando los paréntesis: \(5x + 8 - 9 = 9x - (-7 + 1)\) \(5x - 1 = 9x - (6)\) Ahora procedemos a aislar la variable \(x\): \(5x - 9x = -6 + 1\) \(-4x = -5\) \(x = \frac{-5}{-4}\) \(x = \frac{5}{4}\) Por lo tanto, la solución de la ecuación es \(x = \frac{5}{4}\).

Solving Equations with Identity Result

Para resolver la ecuación proporcionada en la imagen, primero simplificaremos y combinararemos términos semejantes en ambos lados de la ecuación. Aquí está la ecuación original: \[ 3 + 3x - 1 = x + 2 + 2x \] Combinamos los términos que no contienen la incógnita \( x \) y los términos que contienen \( x \) en ambos lados: \[ (3 - 1) + 3x = (2) + (x + 2x) \] Al hacer esto obtenemos: \[ 2 + 3x = 2 + 3x \] Podemos ver que ambos lados de la ecuación son idénticos, lo que significa que la ecuación se cumple para cualquier valor de \( x \). En otras palabras, tenemos una identidad. Esto significa que la ecuación es verdadera para cualquier valor de \( x \) y no es necesario resolver para una valor específico.

Solving Linear Equations

Claro, vamos a resolver la ecuación paso a paso. La ecuación original es: \[ 3 + 3x - 1 = x + 2 + x \] Primero, simplificaremos ambos lados de la ecuación combinando términos semejantes. En el lado izquierdo, \( 3 - 1 \) se simplifica a \( 2 \), quedándonos la ecuación así: \[ 3x + 2 = x + 2 + x \] En el lado derecho, \( x + x \) se simplifica a \( 2x \), entonces la ecuación queda: \[ 3x + 2 = 2x + 2 \] Ahora, restamos \( 2x \) en ambos lados para obtener los términos con \( x \) en un solo lado: \[ 3x + 2 - 2x = 2x + 2 - 2x \] \[ x + 2 = 2 \] Después, restamos \( 2 \) en ambos lados para despejar \( x \): \[ x + 2 - 2 = 2 - 2 \] \[ x = 0 \] Por lo tanto, la solución para \( x \) es 0.

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 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

Давайте решим уравнение, представленное на изображении: \( \frac{5}{x - 3} + 27 = \frac{18}{x - 3} \) Чтобы решить это уравнение, нам нужно изолировать переменную \( x \). Так как оба слагаемых с левой и правой стороны уравнения имеют общий знаменатель \( x - 3 \), мы можем исключить его, умножив все части уравнения на \( x - 3 \), чтобы избавиться от дробей. \( (x - 3) \cdot \frac{5}{x - 3} + 27 \cdot (x - 3) = (x - 3) \cdot \frac{18}{x - 3} \) После упрощения мы получим: \( 5 + 27x - 81 = 18 \) Теперь упростим уравнение, сложив члены: \( 27x - 76 = 18 \) Добавим 76 к обеим сторонам уравнения, чтобы перенести константу с левой стороны на правую: \( 27x = 18 + 76 \) \( 27x = 94 \) Теперь разделим обе стороны уравнения на 27, чтобы найти значение \( x \): \( x = \frac{94}{27} \) Таким образом, решение уравнения: \( x = \frac{94}{27} \)

Binomial Equation Simplification

이 이미지에는 중학교 수학 문제 같습니다. 방정식 시퀀스를 단순화하는 문제입니다. 주어진 식들을 단순화하면 다음과 같습니다. 1. \(a_0C_0 + a_1C_1 + a_2C_2 + \ldots + a_nC_n\) 이 식은 이항 정리의 일반 항에서의 계수와 같은 형태입니다. 여기서 \(a_k\)는 상수일 수 있고, \(C_k\)는 \(n\)개 중 \(k\)를 선택하는 조합, 즉 \(nCk\)를 의미합니다. \(a_k\)가 모든 \(k\)에 대해 \(1\)이라면, 이 식은 단순히 이항정리의 전개 결과 합과 같아져, \((1+1)^n\)이고, 그 결과는 \(2^n\)이 됩니다. 2. \(a_0C_n - a_1C_{n-1} + a_2C_{n-2} + \ldots + a_nC_0\) 이 식은 조금 더 복잡해 보입니다. 이것은 첫 번째 항과 마지막 항의 지수가 서로 반대되어 있고, 부호가 교대로 바뀌고 있습니다. 이는 \((1-1)^n\)의 전개와 비슷하게 보이며, 이 경우 결과는 \(0\)이 됩니다. 왜냐하면 이는 실제로 \(0^n\)과 같으며, \(n>0\)이면 항상 \(0\)이 되기 때문입니다. 3. \(a_0C_0 + a_1C_2 + a_2C_4 + \ldots + a_nC_{2n}\) 이 식은 지수가 다릅니다. 이 경우 식을 단순화하기 위한 명확한 규칙이 보이지 않습니다. 주어진 정보가 더 필요합니다. 예를 들어, \(a_k\)의 값과 \(n\)의 범위에 따라 결과가 달라질 수 있습니다. 위의 분석은 이항정리에 대한 일반적인 지식을 기반으로 한 것입니다. 주어진 \(a_k\)의 값 또는 \(n\)의 구체적인 범위 없이는 정확한 해답을 제공하기 어렵습니다.

Solving Radical Equation for p

To solve the equation involving the radical, we want to isolate p. Given the equation from the image: \[ \sqrt{\frac{2p+1}{t}} = 3 \] First, square both sides of the equation to eliminate the square root: \[ \left( \sqrt{\frac{2p+1}{t}} \right)^2 = 3^2 \] This simplifies to: \[ \frac{2p+1}{t} = 9 \] Next, multiply both sides by t to remove the denominator: \[ 2p+1 = 9t \] Subtract 1 from both sides to isolate the term with p: \[ 2p = 9t - 1 \] Finally, divide both sides by 2 to solve for p: \[ p = \frac{9t - 1}{2} \] Therefore, the value of p in terms of t is: \[ p = \frac{9t - 1}{2} \]