slope of \( \frac{2}{3} \) that passes through \( (6,4) \)

To find the equation of a line with a slope of \( \frac{2}{3} \) that passes through the point \( (6, 4) \), we can use the point-slope form of a linear equation, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Plugging in the values, we get: \[ y - 4 = \frac{2}{3}(x - 6). \] Simplifying this will give us the line's equation. Now, let’s rearrange it to see it in slope-intercept form \( y = mx + b \): 1. Distribute \( \frac{2}{3} \): \[ y - 4 = \frac{2}{3}x - 4 \] 2. Add 4 to both sides: \[ y = \frac{2}{3}x \] So the line equation is: \[ y = \frac{2}{3}x + 0 \] This line has a y-intercept at \( (0, 0) \). If you’re curious about the slopes of other lines, did you know that a slope of 1 indicates a line that rises diagonally at 45 degrees? It’s the perfect diagonal line that creates equal rise and run! On the other hand, understanding lines is fundamental in many fields! Architects and engineers constantly work with slopes when designing buildings, ramps, or roads to ensure safety and functionality. Merging math with the creative design process is where the magic happens!