Verified Answer :

  • Substitute \(x = 8\) into the linear equation \(y = \frac{3}{2}x - 3\).
  • Multiply \(\frac{3}{2}\) by \(8\) to obtain \(12\).
  • Subtract \(3\) from \(12\) to find the final value.
  • The result is \(\boxed{9}\).

Explanation

  1. Problem Analysis and Objective

We are given the linear function \(y = \frac{3}{2}x - 3\). Our objective is to determine the value of \(y\) for the specific input value \(x = 8\). This is a straightforward substitution problem where we replace the variable \(x\) with its given numerical value and simplify the resulting expression.

  1. Substitution and Calculation

To find the value of \(y\) when \(x = 8\), we substitute \(x = 8\) into the given equation:
\[y = \frac{3}{2}(8) - 3\]
Now, we calculate the product of \(\frac{3}{2}\) and \(8\). Dividing \(8\) by \(2\) gives \(4\), and multiplying \(4\) by \(3\) results in \(12\):
\[y = 12 - 3\]
Finally, subtracting \(3\) from \(12\) gives us the result:
\[y = 9\]
Thus, when \(x = 8\), the value of \(y\) is \(9\).

Examples

Linear functions are used in many everyday situations, such as calculating the cost of a service that has a fixed fee and an hourly rate. Suppose a repair person charges a base fee of \(\$3\) and an additional rate of \(\$1.50\) per hour of work. The cost \(y\) after \(x\) hours is given by the function \(y = 1.5x - 3\). If the repair person works for \(8\) hours, the total cost would be \(y = \frac{3}{2}(8) - 3 = \boxed{9}\) dollars. This model helps businesses and customers predict costs accurately.

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