$y=2 x-5$
$x$ | $y$ |
-2 | □ |
1 | □ |
6 | □ |
7 | □ |
The objective is to determine the output values of the linear function \(y = 2x - 5\) for the input values \(x = -2, 1, 6,\) and \(7\). A linear function is a mathematical relationship where each input \(x\) corresponds to exactly one output \(y\), forming a straight line when graphed on a coordinate plane. Here, we evaluate the expression \(2x - 5\) for each of the given values of \(x\) to complete the requested table.
To find the corresponding \(y\) values, we substitute each value of \(x\) into the equation \(y = 2x - 5\).
Based on the calculations performed, we can construct the final table summarizing the relationship between \(x\) and \(y\):
| \(x\) | \(y\) \) \(-2\) | \(-9\) \) \(1\) | \(-3\) \) \(6\) | \(7\) \) \(7\) | \(9\) \) |
This completes the required table for the given linear equation \(\boxed{y = 2x - 5}\).
Linear functions are used in everyday life to calculate costs based on usage. For example, if a taxi service charges a base fee of \(\$5\) and an additional \(\$2\) for every kilometer traveled, the total cost \(y\) for traveling \(x\) kilometers is given by the equation \(y = 2x + 5\) (if we consider adding the fee). Similar linear relationships appear when calculating savings, phone plans, or predicting growth patterns, making this foundational concept essential for modeling real-world situations accurately.