Questions: Solve the system of equations using the elimination method. 4x - 2y = 6 2/3x + 2/3y = 2 One solution: No solution Infinite number of solutions Enter your answer as a point. Example: (2,5) Select no solutions for a system with no solutions. Select infinitely number of solutions for a system with infinitely many solutions.

Solution Steps

To solve the system of equations using the elimination method, we first need to eliminate one of the variables by making the coefficients of either \(x\) or \(y\) the same in both equations. We can achieve this by multiplying the second equation by a suitable number. Once the coefficients are aligned, we subtract one equation from the other to eliminate one variable, solve for the remaining variable, and then substitute back to find the other variable.

We start with the following system of equations: \[ \begin{align*}

1. & \quad 4x - 2y = 6 \\
2. & \quad \frac{2}{3}x + \frac{2}{3}y = 2 \end{align*} \]

To eliminate the fractions in the second equation, we multiply the entire equation by 3: \[ 2x + 2y = 6 \]

Now we have the modified system: \[ \begin{align*}

1. & \quad 4x - 2y = 6 \\
2. & \quad 2x + 2y = 6 \end{align*} \] We can solve this system using elimination. By manipulating the equations, we find:

• From the second equation, we can express \(y\) in terms of \(x\): \[ 2y = 6 - 2x \implies y = 3 - x \]
• Substituting \(y\) into the first equation: \[ 4x - 2(3 - x) = 6 \implies 4x - 6 + 2x = 6 \implies 6x - 6 = 6 \implies 6x = 12 \implies x = 2 \]
• Now substituting \(x = 2\) back into the expression for \(y\): \[ y = 3 - 2 = 1 \]

Final Answer