Question

a system of equations is given.
x + 3y = 1
-2x - y = -2
which system of equations has the same solution as the given system?
a
-2x + 3y = 1
-2x - y = -2
b
2x + 6y = 1
2x + y = 2
c
x + 3y = 1
-x - y = -2
d
2x + 6y = 2
-2x - y = -2

Explanation:

Step1: Analyze the given system

The given system is \(

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

\). We can use the properties of systems of linear equations (like multiplying an equation by a constant or adding/subtracting equations) to find an equivalent system.

Step2: Check option D

For option D, the first equation is \( 2x + 6y = 2 \). Notice that if we multiply the first equation of the given system (\( x + 3y = 1 \)) by 2, we get \( 2(x + 3y)=2(1) \), which simplifies to \( 2x + 6y = 2 \). The second equation in option D is \( -2x - y = -2 \), which is the same as the second equation of the given system. So the system in option D is equivalent (has the same solution) because we just scaled the first equation by 2 (a valid operation that doesn't change the solution set).

Step3: Verify other options (briefly)

• Option A: First equation is \( -2x + 3y = 1 \), which is different from the given first equation (scaling or operation doesn't lead to this from \( x + 3y = 1 \)).
• Option B: First equation \( 2x + 6y = 1 \) (multiplying given first equation by 2 gives \( 2x + 6y = 2 \), not 1), and second equation \( 2x + y = 2 \) (different from \( -2x - y = -2 \)).
• Option C: First equation same as given, but second equation \( -x - y = -2 \) is different from \( -2x - y = -2 \).

Answer:

D. \(

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

\)