Which of the following is an equation of the line in the xy-plane that passes through the points (1, 0) and (0, 5)?
y = -5x + 5 represents the equation of the line passing through the points (1, 0) and (0, 5).
To find the equation of the line through the given points, we calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), which yields a slope of -5. Using the point-slope form, we can derive the equation \( y = -5x + 5 \).
This equation has the correct slope of -5 but does not pass through the point (1, 0). If we substitute x = 1, we get y = -5(1) + 1 = -4, which is not equal to 0. Therefore, it does not represent the line that connects the two points.
This equation accurately describes the line passing through both points. Substituting (1, 0) yields 0 = -5(1) + 5, which is true, and substituting (0, 5) yields 5 = -5(0) + 5, also true. This confirms it as the correct equation.
This equation has a positive slope of 5, which is incorrect since the slope between the points (1, 0) and (0, 5) is negative. Thus, it cannot represent the line connecting these two points.
Similar to option C, this equation also has a positive slope of 5 and does not correspond to the required negative slope. Furthermore, substituting (0, 5) gives 5 = 5(0) + 5, which is true, but it fails to intersect at (1, 0).
The equation of the line that passes through the points (1, 0) and (0, 5) is represented by y = -5x + 5. This equation correctly reflects the negative slope derived from the coordinates of the points, ensuring it accurately describes the relationship between them. The other options presented either do not pass through the required points or have incorrect slopes, confirming the uniqueness of the correct answer.
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