Question

-6x-3y=12
-12x-6y=-24

Answer 1

To solve the system of equations:

1. \( -6x - 3y = 12 \)
2. \( -12x - 6y = -24 \)

We can start by simplifying the second equation. Notice that if we divide the entire second equation by -6, we get:

\[
-12x - 6y = -24 \implies 2x + y = 4
\]

Now we have a simplified version of both equations:

1. \( -6x - 3y = 12 \)
2. \( 2x + y = 4 \)

Next, let's express \( y \) in terms of \( x \) from the second equation:

\[
y = 4 - 2x
\]

Now, we can substitute \( y \) in the first equation:

\[
-6x - 3(4 - 2x) = 12
\]

Now, distribute -3:

\[
-6x - 12 + 6x = 12
\]

Combine like terms:

\[
-12 = 12
\]

This is a contradiction, which indicates that the two equations are actually dependent (they represent the same line). Therefore, there are infinitely many solutions along the line represented by either equation.

To express the solution set, we can use the form of the second equation \( 2x + y = 4 \):

\[
y = 4 - 2x
\]

This means for any value of \( x \), you can find a corresponding \( y \) using this equation, confirming that there are infinitely many solutions.