Question 6 (Multiple Choice Worth 4 points)

The graph of this system of equations is which of the following?

[tex]\[
\begin{array}{l}
2x + y = 6 \\
6x + 3y = 12
\end{array}
\][/tex]

A. Overlapping lines

B. Parallel lines

C. Intersecting lines

D. A curve intersecting with a line



Answer :

To determine the nature of the graph of the given system of equations, let's analyze the two equations.

The equations are:
[tex]\[ \begin{cases} 2x + y = 6 \\ 6x + 3y = 12 \end{cases} \][/tex]

First, we simplify the second equation by dividing every term by 3:
[tex]\[ 6x + 3y = 12 \implies 2x + y = 4 \][/tex]

Now we have the simplified system:
[tex]\[ \begin{cases} 2x + y = 6 \\ 2x + y = 4 \end{cases} \][/tex]

This simplified system clearly shows that the two equations have the same coefficients for x and y, but different constants on the right-hand side (6 and 4).

When two linear equations have the same left-hand side (coefficients for x and y) but different right-hand sides, they represent parallel lines. Parallel lines have the same slope but different intercepts, which means they never intersect.

Therefore, the graph of this system of equations is:

Parallel lines