To determine which statement about the given system of equations is correct, let's analyze the system step by step.
We are given two linear equations:
1) [tex]\(2x - 1 = y\)[/tex]
2) [tex]\(3x - 1 = y\)[/tex]
Since both equations equal [tex]\(y\)[/tex], we can set them equal to each other to find [tex]\(x\)[/tex]:
[tex]\[2x - 1 = 3x - 1\][/tex]
Let's start by isolating [tex]\(x\)[/tex]:
[tex]\[2x - 1 = 3x - 1\][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[-1 = x - 1\][/tex]
Add 1 to both sides:
[tex]\[0 = x\][/tex]
Thus, solving the above, we find:
[tex]\[x = 0\][/tex]
Now that we've found [tex]\(x = 0\)[/tex], substitute [tex]\(x\)[/tex] back into either original equation to find [tex]\(y\)[/tex]. Let's use the first equation:
[tex]\[y = 2(0) - 1\][/tex]
[tex]\[y = -1\][/tex]
So, the solution to the system is:
[tex]\[(x, y) = (0, -1)\][/tex]
Now let’s evaluate the statements:
(A) There is only one [tex]\((x, y)\)[/tex] solution and [tex]\(y\)[/tex] is positive.
- This is incorrect because [tex]\(y = -1\)[/tex] is not positive.
(B) There is only one [tex]\((x, y)\)[/tex] solution and [tex]\(y\)[/tex] is negative.
- This is correct because [tex]\(y = -1\)[/tex] is negative.
(C) There are infinitely many [tex]\((x, y)\)[/tex] solutions.
- This is incorrect because we found exactly one solution.
(D) There are no [tex]\((x, y)\)[/tex] solutions.
- This is incorrect because we found one solution.
Therefore, the correct answer is:
(B) There is only one [tex]\((x, y)\)[/tex] solution and [tex]\(y\)[/tex] is negative.
