Verified Answer :

  • The problem asks to identify which ordered pair satisfies the linear equation \(y = 2x + 1\).
  • By substituting each x-coordinate of the options (0, 1, 2, -1) into the expression \(2x + 1\), we calculate the resulting y-values.
  • Comparing these results, only the pair \((-1, -1)\) satisfies the original equation.
  • The correct ordered pair is \(\boxed{d}\).

Explanation

  1. Analysis of the Problem

To determine which of the provided ordered pairs \((x, y)\) lies on the line defined by the equation \(y = 2x + 1\), we must check if each pair satisfies this relationship. An ordered pair satisfies the equation if, when we substitute its x-coordinate for \(x\), the result is equal to its corresponding y-coordinate.

  1. Evaluating the Candidate Pairs

We evaluate each candidate pair by calculating the value of \(2x + 1\) using the x-coordinate provided:

  • For (0, 3): substitute \(x = 0\), we get \(2(0) + 1 = 1\). Since \(1 eq 3\), this point is not on the line.
  • For (1, 2): substitute \(x = 1\), we get \(2(1) + 1 = 3\). Since \(3 eq 2\), this point is not on the line.
  • For (2, 1): substitute \(x = 2\), we get \(2(2) + 1 = 5\). Since \(5 eq 1\), this point is not on the line.
  • For (-1, -1): substitute \(x = -1\), we get \(2(-1) + 1 = -2 + 1 = -1\). Since \(-1 = -1\), this point satisfies the equation.
  1. Conclusion

By substituting the x-coordinates into the linear equation, we confirmed that only the ordered pair \((-1, -1)\) results in the y-coordinate specified in the pair. Therefore, the ordered pair \((-1, -1)\) is the only point on the line \(y = 2x + 1\). The correct choice is d. The final answer is \(\boxed{d}\).

Examples

Understanding linear equations is essential in everyday scenarios where there's a predictable rate of change. For instance, if a ride-sharing service charges a flat fee of \(\$1\) and an additional cost of \(\$2\) for every mile traveled, the total cost \(y\) for a trip of \(x\) miles can be modeled as \(y = 2x + 1\). By testing the point \((-1, -1)\) which would represent a negative mile traveled, we can identify that this specific model doesn't apply to those negative values, helping us understand the domain and applicability of linear relationships in real-world contexts like pricing, resource allocation, and physics.

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