To write a linear equation in the form \( y = mx + b \) based on the given table, we first need to identify the slope \( m \) and the y-intercept \( b \).
From the table data, we have the following points:
- \((0, 0)\)
- \((4, 4)\)
- \((4, -3)\)
- \((8, 8)\)
- \((-10, -10)\)
It appears that not all points provided are valid, especially since \((4, 4)\) and \((4, -3)\) share the same x-coordinate but have different y-values, which does not conform to a linear relationship. To facilitate finding a linear equation, we can consider the points that adhere to a linear relationship.
Let's take points \((0, 0)\) and \((8, 8)\), as they give a clearer view of a trend.
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Determine the slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 0}{8 - 0} = \frac{8}{8} = 1 \]
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Determine the y-intercept \( b \): From the point \((0, 0)\), when \( x = 0 \), \( y = 0 \). Thus, \( b = 0 \).
Putting this together, the equation becomes: \[ y = 1x + 0 \quad \text{or simply} \quad y = x \]
So, the linear equation based on the valid points that we can gather is: \[ y = x \]