What is an equation of the line that passes through the point \( (5,4) \) and is parallel to the line \( 4 x+5 y=25 \) ?

To find an equation of a line that is parallel to the line \( 4x + 5y = 25 \), we first need the slope of the given line. We can rearrange it into slope-intercept form, \( y = mx + b \). Starting from \( 4x + 5y = 25 \), we isolate \( y \): \[ 5y = -4x + 25 \] \[ y = -\frac{4}{5}x + 5 \] The slope \( m \) of the original line is \( -\frac{4}{5} \). Since parallel lines have the same slope, our new line passing through the point \( (5, 4) \) will also have this slope. We can use the point-slope form of a line equation: \[ y - y_1 = m(x - x_1) \] Plugging in the point \( (5, 4) \) and the slope \( -\frac{4}{5} \): \[ y - 4 = -\frac{4}{5}(x - 5) \] This simplifies to: \[ y - 4 = -\frac{4}{5}x + 4 \] \[ y = -\frac{4}{5}x + 8 \] So, the equation of the line that passes through \( (5, 4) \) and is parallel to \( 4x + 5y = 25 \) is \( y = -\frac{4}{5}x + 8 \). If you ever find yourself unsure about the slope, just swap your line equation to that convenient \( y = mx + b \) form, and voila! You’ve got the slope ready for action, just like preparing a secret weapon for your math battle. And speaking of battles, parallel lines will never cross swords—err, I mean intersect! That’s their secret to maintaining harmony in the math world. They share the same slope, which keeps their paths aligned perfectly, forever side by side.