Question

solve the system of equations \\(\

$$\begin{cases} 2x - 6y = 30 \\\\ y = \\dfrac{1}{3}x + 1 \\end{cases}$$

\\) using the provided coordinate grid (chart) to graph the lines and find the solution (or analyze the system).

Explanation:

Step1: Rewrite the first equation

Rewrite \(2x - 6y = 30\) in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept).
First, isolate \(y\):
\(-6y=-2x + 30\)
Divide each term by \(-6\): \(y=\frac{-2}{-6}x+\frac{30}{-6}\), which simplifies to \(y=\frac{1}{3}x - 5\)

Step2: Analyze the two lines

The second equation is \(y=\frac{1}{3}x + 1\)
For a linear equation \(y = mx + b\), if two lines have the same slope (\(m\)) and different y - intercepts (\(b\)), they are parallel.
The slope of \(y=\frac{1}{3}x - 5\) is \(\frac{1}{3}\) and the slope of \(y=\frac{1}{3}x + 1\) is also \(\frac{1}{3}\), and their y - intercepts (\(-5\) and \(1\)) are different. So the two lines are parallel and will never intersect. This means the system of equations has no solution.

Answer:

The system of equations \(

$$\begin{cases}2x - 6y=30\\y=\frac{1}{3}x + 1\end{cases}$$

\) has no solution because the two lines are parallel (same slope \(\frac{1}{3}\), different y - intercepts).