Questions: Solve the system of equations by elimination/addition. Give your solution as an ordered pair. -9x + 9y = 6 -6x + 6y = 4

Solution Steps

To solve the system of equations by elimination/addition, we need to eliminate one of the variables by adding or subtracting the equations. Here are the steps:

1. Multiply each equation by a suitable number so that the coefficients of one of the variables are opposites.
2. Add or subtract the equations to eliminate that variable.
3. Solve for the remaining variable.
4. Substitute the value of the solved variable back into one of the original equations to find the value of the other variable.
5. Write the solution as an ordered pair.

We are given the following system of equations: \[ \begin{array}{l} -9x + 9y = 6 \\ -6x + 6y = 4 \end{array} \]

First, we can simplify both equations by dividing each term by the greatest common divisor of the coefficients.

For the first equation: \[ -9x + 9y = 6 \quad \Rightarrow \quad -x + y = \frac{6}{9} \quad \Rightarrow \quad -x + y = \frac{2}{3} \]

For the second equation: \[ -6x + 6y = 4 \quad \Rightarrow \quad -x + y = \frac{4}{6} \quad \Rightarrow \quad -x + y = \frac{2}{3} \]

We observe that both simplified equations are identical: \[ -x + y = \frac{2}{3} \]

Since both equations are the same, we can solve for one variable in terms of the other. Let's solve for \( y \): \[ y = x + \frac{2}{3} \]

Since the two equations are identical, the system has infinitely many solutions. The solutions can be expressed as: \[ (x, y) = \left( x, x + \frac{2}{3} \right) \]

Final Answer