Use the table to answer the question.

Question

x-value y-value if y=2x−5 y-value if y=x−3
0 −5 −3
1 −3 −2
2 −1 −1
3 1 0
Jian made a table showing some x- and y-values in a system of equations. Use his table to identify the solution for the system.

(1 point)
Responses

There is an infinite number of solutions.
There is an infinite number of solutions.

There is no solution.
There is no solution.

(3,0)
left parenthesis 3 comma 0 right parenthesis

(2,−1)
left parenthesis 2 comma negative 1 right parenthesis
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Answer 1

To find the solution of the system of equations given the two equations represented by the y-values, we will look for a common (x, y) pair in the two sets of y-values.

From the table:

• For \(y = 2x - 5\):

  • When \(x = 0\), \(y = -5\)
  • When \(x = 1\), \(y = -3\)
  • When \(x = 2\), \(y = -1\)
  • When \(x = 3\), \(y = 1\)

• For \(y = x - 3\):

  • When \(x = 0\), \(y = -3\)
  • When \(x = 1\), \(y = -2\)
  • When \(x = 2\), \(y = -1\)
  • When \(x = 3\), \(y = 0\)

Now, let's compare the y-values from both equations for each x-value:

• At \(x = 0\): \(y = -5\) (from \(y = 2x - 5\)) and \(y = -3\) (from \(y = x - 3\)) — not a solution.
• At \(x = 1\): \(y = -3\) and \(y = -2\) — not a solution.
• At \(x = 2\): \(y = -1\) and \(y = -1\) — this is a solution! \((2, -1)\)
• At \(x = 3\): \(y = 1\) and \(y = 0\) — not a solution.

The pair \((2, -1)\) is the only point where both equations meet, thus it is the solution to the system of equations.

Therefore, the answer is: \((2, -1)\)