Example Question - no solution
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Determining the Constant Value for a System of Equations Without Solutions
The system of equations is given by: \[ \begin{align*} 3x + \frac{1}{2}y &= \frac{3}{2} \\ 2x + \frac{3}{4}y &= 1 \\ \frac{1}{2}x + \frac{1}{4}y &= \frac{p}{2} + \frac{1}{2} \end{align*} \] To determine a value of \( p \) that leads to no solution, the equations must be inconsistent. This happens when the ratios of the coefficients of \( x \) and \( y \) are equal for two equations, but the ratio of the constants is different. First, let's make the coefficients of \( y \) similar in the first two equations: \[ \begin{align*} 3x + \frac{1}{2}y &= \frac{3}{2} \ |\times2 \\ 2x + \frac{3}{4}y &= 1 \ |\times4 \\ \end{align*} \] This results in: \[ \begin{align*} 6x + y &= 3 \\ 8x + 3y &= 4 \end{align*} \] Now, let's make the coefficient of \( x \) in the third equation similar to the first equation: \[ \begin{align*} \frac{1}{2}x + \frac{1}{4}y &= \frac{p}{2} + \frac{1}{2} \ |\times6 \\ \end{align*} \] This results in: \[ 3x + \frac{3}{2}y = 3p + 3 \] Comparing the first and last equations, the coefficient ratio of \( x \) to \( y \) is \( 6:1 \) and \( 3:\frac{3}{2} \). Simplifying the second ratio gives us \( 2:1 \), which is the same as \( 6:3 \). For the system to have no solution, the constant term (after the simplification) should not have the same ratio. Comparing the constants from the first and the last equations, \( 3 \) and \( 3p + 3 \) must not have the same ratio as the coefficients of \( x \) and \( y \), which is \( 2:1 \). Therefore, \( 2(3) \) should not equal \( 3p + 3 \). Equating and solving gives us: \[ 6 \neq 3p + 3 \] Subtracting \( 3 \) from both sides gives: \[ 3 \neq 3p \] Dividing by \( 3 \) we get: \[ 1 \neq p \] Thus, the value of \( p \) that results in no solution is any value except \( p = 1 \).
Solving an Inconsistent Equation
Claro, voy a resolver la ecuación paso por paso. La ecuación dada es: 5 + 2(1 - x) = -2x + 3 Comenzaremos distribuyendo el 2 que está multiplicando al paréntesis en la parte izquierda de la ecuación: 5 + 2*1 - 2*x = -2x + 3 5 + 2 - 2x = -2x + 3 Ahora simplificamos sumando 5 y 2 en el lado izquierdo: 7 - 2x = -2x + 3 Después queremos deshacernos de los términos con x en uno de los lados. Sin embargo, observamos que el coeficiente de x es igual en ambos lados de la ecuación (-2x), por lo que al intentar eliminar x, estos términos se cancelarían mutuamente. Prosigamos y veamos qué sucede: Si sumamos 2x a ambos lados para intentar deshacernos de los x, obtenemos: 7 - 2x + 2x = -2x + 2x + 3 7 = 3 Esto nos lleva a una contradicción porque sabemos que 7 no es igual a 3. Esto indica que la ecuación original no tiene solución. En otras palabras, no hay un valor de x que satisfaga la ecuación inicial. Entonces, la ecuación no tiene solución, lo que significa que es una ecuación inconsistente.
System of Linear Equations with No Solution
The image shows a system of linear equations: 3x − y = 3 3x − y = 4 To determine the solution to the system, we can compare the two equations directly. Both equations have the same coefficients for x and y, but different constant terms (3 and 4, respectively). This means that the two lines represented by these equations are parallel. Because they are parallel, they never intersect, which means there is no point (x,y) that satisfies both equations simultaneously. Therefore, the system has no solution. Checking the answer options, the correct choice would be: (B) The system has no solution.
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