Writing the equation of a line through two given points Find an equation for the line that passes through the points \( (5,2) \) and \( (1,4) \).
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To find the equation of the line passing through the points \( (5,2) \) and \( (1,4) \), we first calculate the slope \( m \) using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging in our points: \( m = \frac{4 - 2}{1 - 5} = \frac{2}{-4} = -\frac{1}{2} \). Now, using the slope-intercept form \( y = mx + b \), we can choose one of the points to solve for \( b \). Substituting point \( (5,2) \): \[ 2 = -\frac{1}{2}(5) + b \] Solving gives us \( b = 2 + \frac{5}{2} = \frac{9}{2} \). Therefore, the equation of the line is \[ y = -\frac{1}{2}x + \frac{9}{2} \]. Remember, writing it in standard form is also a great option for a neat appearance! If you're working with coordinates, it can be helpful to visualize those points on a graph. Plotting \( (5,2) \) and \( (1,4) \) not only helps you understand the relationship between them but also provides a visual confirmation that your line accurately represents the two points. Visual aids like graphs can turn abstract equations into tangible relationships, making the learning process more engaging!
