Which of the following equations defines a line that is parallel to the line \( y=-\frac{4}{3} x-4 \) and passes through the point \( (3,-1) \) ?

To find a line that is parallel to the line \( y = -\frac{4}{3} x - 4 \), we know that parallel lines share the same slope. The slope here is \( -\frac{4}{3} \). Now, using the point-slope form of a linear equation, which is \( y - y_1 = m(x - x_1) \), we can substitute \( m = -\frac{4}{3} \) and the point \( (3, -1) \): \[ y - (-1) = -\frac{4}{3}(x - 3) \] This simplifies to: \[ y + 1 = -\frac{4}{3}x + 4 \] \[ y = -\frac{4}{3}x + 3 \] So the equation of the desired parallel line is \( y = -\frac{4}{3}x + 3 \). To find where parallel lines fit into the larger picture, it's interesting to note that parallel lines maintain a constant distance from each other, forming important constructs in geometry like shapes and angles. Artists often use parallel lines to create depth and perspective, making them essential in both mathematical expressions and creative designs. When writing equations for lines, a common mistake is forgetting to use the slope-intercept form consistently or miscalculating the slope! Always double-check your slope calculations by ensuring you correctly identify the changes in \( y \) and \( x \) from the two points you’re using, and remember to follow through with the right format as you derive your line's equation.