Verified Answer :

  • Assign the values \((x, y) = (5, 1)\) to the given inequality \(15x + y > 7\).
  • Calculate the expression: \(15(5) + 1 = 75 + 1 = 76\).
  • Compare the result to the threshold: \(76 > 7\).
  • The point satisfies the inequality, so the answer is \(\boxed{yes}\).

Explanation

  1. Analysis of the Problem

The problem asks whether the point \((5, 1)\) satisfies the linear inequality \(15x + y > 7\). To determine this, we need to substitute the \(x\)-coordinate and the \(y\)-coordinate of the given point into the left-hand side of the inequality and check if the resulting value is strictly greater than \(7\).

  1. Substitution and Calculation

We are given the point \((x, y) = (5, 1)\). Substituting \(x = 5\) and \(y = 1\) into the expression \(15x + y\), we calculate the value as follows:
\[15(5) + 1 = 75 + 1 = 76\]

  1. Comparison and Conclusion

Now we compare the calculated value, \(76\), to the value given in the inequality, which is \(7\). We must verify if \(76 > 7\). Since \(76\) is clearly greater than \(7\), the inequality holds true. Therefore, the point \((5, 1)\) satisfies the given inequality.

Examples

In real life, linear inequalities are often used in resource allocation or budgeting. For example, if you have a budget of \$7 (representing the threshold) and you are analyzing a purchase where each item of type A costs \$15 and each item of type B costs \$1, buying 5 of A and 1 of B would cost \(\$76\). Since \(\$76\) exceeds your budget of \(\$7\), this choice is not within your budget limits. Similarly, this check confirms whether a specific combination of choices meets or exceeds a defined requirement.

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