Question

1. \

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

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).
Subtract \(2x\) from both sides: \(-6y=-2x + 30\).
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 equations

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.
Here, the slope of the first line \(y=\frac{1}{3}x - 5\) is \(\frac{1}{3}\), and the slope of the second line \(y=\frac{1}{3}x + 1\) is also \(\frac{1}{3}\), and their y - intercepts (\(-5\) and \(1\)) are different.

Step3: Determine the solution of the system

A system of linear equations represents two lines. If two lines are parallel (same slope, different y - intercepts), they never intersect. So, the system of equations \(

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

\) has no solution.

Answer:

The system of equations has no solution.