To find the equation of a non-proportional linear relationship given the points \((-2, -1)\), \((0, 1)\), and \((2, 3)\), we first need to determine the slope of the line using any two of the points.
Let's use the points \((0, 1)\) and \((2, 3)\):
-
Calculate the slope (m): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{2 - 0} = \frac{2}{2} = 1 \]
-
Use the slope-intercept form \(y = mx + b\) to find the equation: Given \(m = 1\), we can substitute \(x = 0\) and \(y = 1\) (from point \((0, 1)\)) to find \(b\): \[ 1 = 1(0) + b \implies b = 1 \]
-
Write the equation: The equation of the line is: \[ y = 1x + 1 \quad \text{or simply} \quad y = x + 1 \]
-
Verify with other points: For \((-2, -1)\): \[ y = -2 + 1 = -1 \quad \text{(This point is on the line)} \] For \((2, 3)\): \[ y = 2 + 1 = 3 \quad \text{(This point is also on the line)} \]
Thus, the equation representing the non-proportional linear relationship of the given coordinates is: \[ \boxed{y = x + 1} \]