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\( x-y=2 \) en \( 2 x+y=10 \)

Ask by Reid Lowe. in South Africa
Feb 20,2025

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Answer

\( x=4 \) and \( y=2 \).

Solution

Solve the system of equations \( x-y=2;2x+y=10 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}x-y=2\\2x+y=10\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}x=2+y\\2x+y=10\end{array}\right.\) - step2: Substitute the value of \(x:\) \(2\left(2+y\right)+y=10\) - step3: Simplify: \(4+3y=10\) - step4: Move the constant to the right side: \(3y=10-4\) - step5: Subtract the numbers: \(3y=6\) - step6: Divide both sides: \(\frac{3y}{3}=\frac{6}{3}\) - step7: Divide the numbers: \(y=2\) - step8: Substitute the value of \(y:\) \(x=2+2\) - step9: Calculate: \(x=4\) - step10: Calculate: \(\left\{ \begin{array}{l}x=4\\y=2\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=4\\y=2\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(4,2\right)\) The solution to the system of equations \(x-y=2\) and \(2x+y=10\) is \(x=4\) and \(y=2\).

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Para resolver el sistema de ecuaciones \( x - y = 2 \) y \( 2x + y = 10 \), podemos usar el método de sustitución o el de eliminación. Despejando \( y \) de la primera ecuación obtenemos \( y = x - 2 \). Luego, sustituimos esta expresión en la segunda ecuación: \( 2x + (x - 2) = 10 \). Esto se simplifica a \( 3x - 2 = 10 \), y al resolver para \( x \), encontramos que \( x = 4 \). Sustituyendo \( x \) en \( y = x - 2 \) nos da \( y = 2 \). ¡Y voilà! La solución es \( (4, 2) \). Este tipo de problemas no solo es útil en el aula, sino que también aparece en situaciones de la vida real, como en la planificación de recursos, presupuestos o en la ingeniería al analizar sistemas. ¡Así que la próxima vez que hagas un presupuesto o planifiques un proyecto, piensa en cómo este tipo de ecuaciones puede ayudarte a optimizar resultados!

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