Write an equation of the line that passes through the points in the table.
$x$ | 0 | 1 | 2 | 3 |
$y$ | 10 | 7 | 4 | 1 |
We are given a set of data points: \((0, 10)\), \((1, 7)\), \((2, 4)\), and \((3, 1)\). We want to find the equation of a line \(y = mx + b\) that passes through all these points. A line is uniquely determined by two points, and we can use any two of these points to find the slope \(m\) and the y-intercept \(b\).
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Using the points \((0, 10)\) and \((1, 7)\):
\[m = \frac{7 - 10}{1 - 0} = \frac{-3}{1} = -3\]
Thus, the slope of the line is \(-3\).
The y-intercept \(b\) occurs where the line crosses the y-axis, which is the point where \(x = 0\). Looking at our data points, we have the point \((0, 10)\). Therefore, when \(x = 0\), \(y = 10\), which means:
\[10 = m(0) + b \implies b = 10\]
So, the y-intercept is \(10\).
Now that we have the slope \(m = -3\) and the y-intercept \(b = 10\), we can construct the equation of the line in the form \(y = mx + b\):
\[y = -3x + 10\]
To verify this, we can check the other points. For \((2, 4)\):
\[y = -3(2) + 10 = -6 + 10 = 4\]
For \((3, 1)\):
\[y = -3(3) + 10 = -9 + 10 = 1\]
Since the equation holds true for all given points, the final equation is \(y = -3x + 10\), so \(y = \boxed{-3x + 10}\).
Linear relationships are fundamental in modeling real-world processes that happen at a constant rate. For example, if you have a battery with a charge of 100% and it loses 10% of its capacity every hour, you can model the remaining charge \(y\) after \(x\) hours using a linear equation \(y = -10x + 100\). Understanding these relationships helps in predicting future states, such as when your device will run out of power, simply by knowing the starting amount and the constant rate of change.