Find an equation of the line containing the given point and having the given slope. \( (0,2), \mathrm{m}=-4 \)

To find the equation of a line when you know a point and the slope, you can use the point-slope form of the line's equation, which is \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) is the point \( (0, 2) \), and \( m \) is the slope, \( -4 \). Plugging these values in, you get: \( y - 2 = -4(x - 0) \), which simplifies to \( y - 2 = -4x \) or, rearranging, \( y = -4x + 2 \). Now you have the equation: \( y = -4x + 2 \)! In everything from economics to engineering, the equation of a line is crucial. For instance, in business, evaluating cost versus sales can often be modeled with linear equations, helping decision-makers optimize pricing strategies. Imagine being able to graphically represent how increasing production affects profits or losses in an engaging way! The equation you just found can also help in understanding relationships between variables. Ever heard of intercepts? They’re your x-intercept and y-intercept. In your case, for the line \( y = -4x + 2 \), the y-intercept is \( 2 \) (where the line crosses the y-axis) and the x-intercept can be found by setting \( y = 0 \) and solving for \( x \). Grab a graphing tool, plot it, and visualize how the line behaves—it’s educational and super fun!