Question

drag each tile to the correct box. arrange the systems of equations in order from least to greatest based on the number of solutions for each system. \\(\

$$\begin{cases}3x - 7y = 9\\\\-4x + 5y = 1\\end{cases}$$

\\) \\(\

$$\begin{cases}-5x + y = 10\\\\-25x + 5y = 50\\end{cases}$$

\\) \\(\

$$\begin{cases}y = 6x - 2\\\\y = 6x - 4\\end{cases}$$

\\)

Explanation:

Step1: Analyze System 1

System:

$$\begin{cases}3x - 7y = 9 \\ -4x + 5y = 1\end{cases}$$

Check the ratio of coefficients:
$\frac{3}{-4}
eq \frac{-7}{5}$, so the lines intersect. Number of solutions: 1.

Step2: Analyze System 2

System:

$$\begin{cases}-5x + y = 10 \\ -25x + 5y = 50\end{cases}$$

Simplify the second equation: divide by 5, get $-5x + y = 10$, which is identical to the first equation. Number of solutions: Infinite.

Step3: Analyze System 3

System:

$$\begin{cases}y = 6x - 2 \\ y = 6x - 4\end{cases}$$

The lines have the same slope ($m=6$) but different y-intercepts, so they are parallel and never intersect. Number of solutions: 0.

Step4: Order by solution count

From least to greatest: 0 solutions, 1 solution, Infinite solutions.

Answer:

1. $\boldsymbol{

$$\begin{cases}y = 6x - 2 \\ y = 6x - 4\end{cases}$$

}$

1. $\boldsymbol{

$$\begin{cases}3x - 7y = 9 \\ -4x + 5y = 1\end{cases}$$

}$

1. $\boldsymbol{

$$\begin{cases}-5x + y = 10 \\ -25x + 5y = 50\end{cases}$$

}$