Example Question - combining like terms
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Solving Linear Equations by Combining Like Terms
<p>\text{Учитывая уравнение: } 2x - 3y = \frac{2}{5}(x - 2y) + \frac{3}{2}(x - y)</p> <p>\text{Первый шаг - избавиться от знаменателей, умножив каждый член на 10 (наименьшее общее кратное знаменателей 5 и 2): }</p> <p>10 \cdot 2x - 10 \cdot 3y = 10 \cdot \frac{2}{5}(x - 2y) + 10 \cdot \frac{3}{2}(x - y)</p> <p>20x - 30y = 4(x - 2y) + 15(x - y)</p> <p>\text{Раскроем скобки: }</p> <p>20x - 30y = 4x - 8y + 15x - 15y</p> <p>\text{Сложим подобные слагаемые справа: }</p> <p>20x - 30y = (4x + 15x) - (8y + 15y)</p> <p>20x - 30y = 19x - 23y</p> <p>\text{Теперь перенесем все члены с переменными на одну сторону, а числа - на другую: }</p> <p>20x - 19x = 23y - 30y</p> <p>x = -7y</p> <p>\text{Финальное уравнение в простейшей форме: } x = -7y.</p>
Expanding and Simplifying Algebraic Expressions
3p(p - q) - (2p - q)² Step 1: Distribute 3p in the first term = 3p² - 3pq Step 2: Expand the square in the second term = (2p - q)(2p - q) = 4p² - 2pq - 2pq + q² Step 3: Combine like terms in the second term = 4p² - 4pq + q² Step 4: Subtract the expanded second term from the first term = (3p² - 3pq) - (4p² - 4pq + q²) Step 5: Distribute the subtraction across each term in the parentheses = 3p² - 3pq - 4p² + 4pq - q² Step 6: Combine like terms = -p² + pq - q² The final answer: = -p² + pq - q²
Simplifying an Algebraic Expression
To simplify the expression \( 3p(p - q) - (2p - q)^2 \), follow these steps: Step 1: Distribute the \(3p\) in the first term: \[ 3p^2 - 3pq \] Step 2: Expand the squared term \((2p - q)^2\): \[ (2p - q)(2p - q) = 4p^2 - 4pq + q^2 \] Step 3: Subtract the expanded squared term from the first term: \[ (3p^2 - 3pq) - (4p^2 - 4pq + q^2) \] Step 4: Distribute the negative sign across the terms in the parentheses: \[ 3p^2 - 3pq - 4p^2 + 4pq - q^2 \] Step 5: Combine like terms: \[ (3p^2 - 4p^2) + (-3pq + 4pq) - q^2 \] \[ -p^2 + pq - q^2 \] So, \( 3p(p - q) - (2p - q)^2 = -p^2 + pq - q^2 \).
Solving Simple Linear Equation
Para resolver la ecuación que se muestra en la imagen, la cual es: \[ \frac{5}{7} = x - 2x \] Primero, buscaremos combinar los términos con x en un lado de la ecuación. Recordemos que \( x - 2x \) es lo mismo que \( 1x - 2x \), lo cual se simplifica a \( -1x \) porque estamos restando 2 veces la x de la x original. Ahora la ecuación se ve así: \[ \frac{5}{7} = -1x \] Para despejar \( x \), necesitamos quitar el coeficiente de -1 que está multiplicando a \( x \). Para hacer eso, podemos dividir ambos lados de la ecuación por -1. \[ -\frac{5}{7} = x \] Entonces, la solución a la ecuación es: \[ x = -\frac{5}{7} \] Hemos encontrado el valor de \( x \) que satisface la ecuación original.
Solving Linear Equation by Eliminating Denominator
Vamos a resolver la ecuación que proporciona la imagen: \( 6 + \frac{7-2x}{5} = 4x + 3 \) Comenzaremos eliminando el denominador para simplificar la ecuación. Para hacerlo, multiplicaremos ambos lados de la ecuación por 5: \( 5 \times (6) + 5 \times \left(\frac{7-2x}{5}\right) = 5 \times (4x) + 5 \times (3) \) Al hacer esto, los términos de la fracción se simplifican: \( 30 + (7 - 2x) = 20x + 15 \) Ahora, simplifica y combina términos semejantes: \( 30 + 7 - 2x = 20x + 15 \) \( 37 - 2x = 20x + 15 \) Lleva las variables para el mismo lado de la ecuación sumando \(2x\) en ambos lados: \( 37 = 22x + 15 \) Ahora, aísle la variable x, restando 15 de ambos lados de la ecuación: \( 37 - 15 = 22x \) \( 22 = 22x \) Finalmente, divida ambos lados por 22 para encontrar el valor de x: \( x = \frac{22}{22} \) \( x = 1 \) La solución a la ecuación es \( x = 1 \).
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 an Algebraic Equation
La ecuación de la imagen es: \[ -5y \cdot (3 + 5) = 10 \] Para resolverla, primero debes multiplicar \( -5y \) por cada término dentro del paréntesis: \[ -5y \cdot 3 + (-5y) \cdot 5 = 10 \] \[ -15y - 25y = 10 \] Ahora, combinamos los términos semejantes: \[ -40y = 10 \] Para encontrar el valor de \( y \), dividimos ambos lados de la ecuación por \( -40 \): \[ y = \frac{10}{-40} \] \[ y = -\frac{1}{4} \] Por lo tanto, la solución es \( y = -\frac{1}{4} \) o \( y = -0.25 \).
Step-by-Step Solution to Linear Equation
Voy a ayudarte a resolver la ecuación paso a paso. La ecuación original que aparece en la imagen es: \[ 3x + 6x - 7 = 82 - 36 \] Primero, combinamos términos semejantes en ambos lados de la ecuación: \[ 3x + 6x = 9x \] \[ 82 - 36 = 46 \] Ahora la ecuación se ve así: \[ 9x - 7 = 46 \] El siguiente paso es despejar \(9x\), sumando 7 en ambos lados de la ecuación para cancelar el -7: \[ 9x - 7 + 7 = 46 + 7 \] \[ 9x = 53 \] Finalmente, dividimos ambos lados de la ecuación por 9 para encontrar el valor de \(x\): \[ x = \frac{53}{9} \] El número 53 no es divisible exactamente por 9, así que esta es la forma simplificada de la fracción. Si necesitas la aproximación decimal de \(x\), seria aproximadamente: \[ x \approx 5.889 \] Por lo tanto, el valor de \(x\) es \(\frac{53}{9}\) o aproximadamente 5.889.
Solving an Equation by Combining Like Terms
To solve the equation \( x + 5 + 6 = 2^3 \), let's first simplify the equation by combining like terms and then solve for \( x \): 1. Combine the constant terms on the left side: \( x + 11 = 2^3 \) 2. Calculate the value of \( 2^3 \): \( 2^3 = 2 \times 2 \times 2 = 8 \) So the equation becomes: \( x + 11 = 8 \) 3. Subtract 11 from both sides to isolate \( x \): \( x + 11 - 11 = 8 - 11 \) \( x = -3 \) 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.
Algebraic Expressions Simplification
Claro, vamos a reducir las siguientes expresiones algebraicas una por una en español: Para la expresión E: \[ E = 4xy^4 - 16xy^3 + xy^4 - 2xy^3 \] Para combinar términos semejantes (aquellos que tienen las mismas variables elevadas a los mismos exponentes), sumamos los coeficientes de estos términos: Los términos \(4xy^4\) y \(xy^4\) son semejantes, por lo que los sumamos: \[ 4xy^4 + xy^4 = 5xy^4 \] Ahora, los términos \(-16xy^3\) y \(-2xy^3\) también son semejantes, por lo que los sumamos: \[ -16xy^3 - 2xy^3 = -18xy^3 \] Entonces, al reducir la expresión E, obtenemos: \[ E = 5xy^4 - 18xy^3 \] Ahora, para la expresión F: \[ F = 14p + 32q + 18q - 15p - 3q \] Los términos \(14p\) y \(-15p\) son semejantes: \[ 14p - 15p = -p \] Los términos \(32q\), \(18q\), y \(-3q\) son semejantes: \[ 32q + 18q - 3q = 47q \] Entonces, al reducir la expresión F, obtenemos: \[ F = -p + 47q \] Finalmente, para la expresión G: \[ G = 13ab^2 + 5ab^2 + ab^2 + 2ab^2 - 6ab^2 \] Todos estos términos son semejantes, así que los sumamos: \[ 13ab^2 + 5ab^2 + ab^2 + 2ab^2 - 6ab^2 = 15ab^2 \] Entonces, al reducir la expresión G, obtenemos: \[ G = 15ab^2 \] Estas son las expresiones reducidas para E, F y G.
Solving a Linear Equation with One Variable
The equation in the image is "2x + 5x + 7 = 0". To solve it, you should first combine the like terms (the terms with x). When you do that: 2x + 5x = 7x So, the equation then becomes: 7x + 7 = 0 Now, to find the value of x, you will move the constant term (7) to the other side of the equation by subtracting it from both sides: 7x = -7 Next, divide both sides of the equation by 7 to isolate x: x = -7 / 7 This simplifies to: x = -1 So the solution to the equation is x = -1.
Solving Expression using Distributive Property and Combining Like Terms
To solve the expression given in the image, you'll need to apply the distributive property and combine the like terms. The expression is: \(-6x^2(3x^5)\) When multiplying two exponential expressions with the same base, you add the exponents: \(x^2 \cdot x^5 = x^{2+5} = x^7\) Now multiply the coefficients, which are the numerical parts: \(-6 \cdot 3 = -18\) Combining both parts, you get: \(-18x^7\) So the simplified expression is: \(-18x^7\)
Solving an Equation with Fractions and Variables
The equation in the image is: \[ \frac{1}{2} - (8x - 10) + 8 = 2(2x + 2) + 9 \] First, we should distribute the negative sign through the parentheses on the left side of the equation and distribute the 2 through the parentheses on the right side. This will give us: \[ \frac{1}{2} - 8x + 10 + 8 = 4x + 4 + 9 \] Next, combine like terms on each side: \[ \frac{1}{2} + 10 + 8 = 18 + \frac{1}{2} \] \[ 18 + \frac{1}{2} - 8x = 4x + 13 \] Combine the numerical terms on the left side: \[ 18.5 - 8x = 4x + 13 \] Next, move all terms containing \( x \) to one side and constant terms to the other side: \[ 18.5 - 13 = 4x + 8x \] \[ 5.5 = 12x \] Finally, divide both sides by 12 to solve for \( x \): \[ x = \frac{5.5}{12} \] \[ x = \frac{11}{24} \] So, \( x = \frac{11}{24} \).
Understanding Equation Solving With Like Terms
To solve the equation \( 403s + 1002s = 1410s \), we can combine like terms on the left side by adding the coefficients of \( s \) together. Here's how you do it step by step: 1. Combine like terms (the terms with \( s \)): \[ 403s + 1002s = (403 + 1002)s = 1405s \] 2. Now that we have \( 1405s \) on the left side and \( 1410s \) on the right side, we set them equal to each other: \[ 1405s = 1410s \] 3. In order to find the solution for \( s \), you generally would isolate \( s \) by dividing both sides of the equation by the coefficient of \( s \). However, since the coefficients of \( s \) are equal and there is no constant term, this would result in an identity where any value of \( s \) satisfies the equation. But since the coefficients are actually different (1405 vs. 1410), there seems to be a mistake in the original problem or a transcription error. If the equation were correctly stated as \( 1405s = 1410s \), the only solution would be \( s = 0 \) because that would be the only way the two sides could be equal. Any non-zero value of \( s \) would result in the two sides not being equal. Since the equation you provided is already balanced with \( 403s + 1002s = 1410s \), which simplifies to \( 1405s = 1410s \), this equation has no solution for \( s \) unless there is a mistake in the terms provided. If the equation is correct, then the solution is that there is no value for \( s \) that would satisfy this equation.
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