What is the equation for a line with a slope of 3 that passes thru (4,7)?

Slope is rise over run. Run is horizontal distance, left to right. Run is always positive because we always go left to right. Rise is the vertical change in that same distance. A negative rise means it drops.

The equation of a line is y = mx where m is the slope. That line passes through the origin. If you want it to pass through some point (a, b) you subtract the coordinates like this: y - b = m(x - a). That is the point-slope form, and you can rewrite it in other forms if it is convenient. You may do this with any point on the line. They all reduce to the same equation.

Given m = 3 and (a, b) = (4, 7) then:

y - 7 = 3(x - 4) point-slope form

y - 7 = 3x - 12

y = 3x - 5 slope-intercept form (intercept is the value of y when x = 0, where the line crosses the vertical axis)

-3x + y + 5 = 0 standard form, all terms on the left, no fractions, and zero or a constant on the right