Pregunta
Determine the number of solutions the system has. \( \begin{array}{l}y+3 x=4 \\ 2 y=-6 x+8 \\ \text { Select Choice }\end{array} \)
Ask by Haynes Estrada. in the United States
Mar 18,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The system has infinitely many solutions.
Solución
Solve the system of equations \( y+3x=4;2y=-6x+8 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}y+3x=4\\2y=-6x+8\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}y=4-3x\\2y=-6x+8\end{array}\right.\)
- step2: Substitute the value of \(y:\)
\(2\left(4-3x\right)=-6x+8\)
- step3: Calculate:
\(\left(x,y\right) = \left(x,4-3x\right),x \in \mathbb{R}\)
- step4: Alternative Form:
\(\textrm{Infinitely many solutions}\)
The system of equations has infinitely many solutions.
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Simplifique esta solución Extra Insights
To determine the number of solutions for the given system of equations, we can manipulate the equations into a more standard form. The first equation is already in a manageable form: \( y + 3x = 4 \) This can be rewritten as \( y = -3x + 4 \). For the second equation, we have: \( 2y = -6x + 8 \) Dividing by 2, we get: \( y = -3x + 4 \). Now we see that both equations simplify to the same line, which means they represent the same set of solutions. In this case, the system has an infinite number of solutions, as every point on the line described by \( y = -3x + 4 \) is a solution to the system.
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