Questions: Does the system have one solution, no solution, or an infinite number of solutions? 3x = y + 3 6x - 2y = 3 Select one: A. One Solution B. Infinite number of solutions C. No solutions (parallel)

Solution Steps

To determine the number of solutions for the given system of linear equations, we can use the method of substitution or elimination. Here, we will use substitution to solve for one variable and then check if the resulting equations are consistent.

We start with the given system of linear equations: \[ \begin{array}{l} 3x = y + 3 \\ 6x - 2y = 3 \end{array} \]

From the first equation, solve for \( y \): \[ 3x = y + 3 \implies y = 3x - 3 \]

Substitute \( y = 3x - 3 \) into the second equation: \[ 6x - 2(3x - 3) = 3 \] Simplify the equation: \[ 6x - 6x + 6 = 3 \implies 6 = 3 \]

The resulting equation \( 6 = 3 \) is a contradiction, which means the system of equations has no solutions. The lines represented by the equations are parallel and do not intersect.

Final Answer