Linear equations model many real-life relationships. For some examples, look at the CBL experiments in the "Real-World Math with the CBL System" book. This is available from most suppliers of TI calculators and materials. For example, my class just did the experiment where then measured the pressure at different depths in a pool. After plotting these points, we could see that the relationship was linear. We could get the equation for this graph and then extrapolate. While we went to only 6 ft., we could use the information to determine the pressure at several 100 ft. down.
Another experiment has students walking to match a given graph. If they have a slanted line, then they must walk at a constant speed -- which gives the relationship between velocity and distance. If the distance from an object is to increase in a linear relationship, then the speed must be constant.
I hope this helps somewhat. Most math books have "problems" that involve linear equations. And any linear equation, such as 3x+2=7, can be solved by graphing y = 3x+2 and y = 7. This is especially useful for very complex equations. Even quadratic equations can be solved this way. For example, to solve 3x^2 - 2x +7 = 0, first get 3x^2 = 2x - 7, then x^2 = 2/3 x - 7/3. If you have a graph of x^2, graph the line y = 2/3 x - 7/3 and locate where they intersect. The intersection is the solution to the quadratic equation -- without having to know how to factor or use the quadratic formula. There is also linear programming, which uses linear inequalities, to solve problems of maximizing profit (or something).