Question

7 what are the solutions to the following system? \

$$\begin{cases} x - 3y = 9 \\\\ -x + 3y = -9 \\end{cases}$$

a infinitely many solutions b (-5, 6) c (5, -6) d no solutions

Explanation:

Step1: Analyze the system of equations

We have the system:

$$\begin{cases} x - 3y = 9 \\ -x + 3y = -9 \end{cases}$$

Notice that if we add the two equations together, we can simplify.

Step2: Add the two equations

Adding the left - hand sides: \((x - 3y)+(-x + 3y)=x - 3y - x + 3y = 0\)
Adding the right - hand sides: \(9+(-9)=9 - 9 = 0\)
So we get \(0 = 0\) after adding the two equations. This means that the two equations are actually the same line (the second equation is just the first equation multiplied by - 1). When two equations represent the same line, there are infinitely many solutions because every point on the line is a solution to both equations.

Answer:

A. infinitely many solutions