How do you determine whether each equation is a linear equation: 5x=y-4?

Look at the $x$ terms and if they are all 1 set of $x$ (no ${x}^{2}$ or anything) then it's linear.
A linear equation will give only one value of $x$ for every value of y. Quadratics or polynomials will more than one power on the $x$ eg. ${x}^{4}$ ) They graph as a wiggly line or a curve.
Linear graphs are straight. Your equation, $5 x = y - 4$ or $y = 5 x + 4$ will be a straight line with a gradient (slope) of 5 and a y-intercept of 4 (where the line cuts through the y axis).