One of the realities of life is how so much of the world runs by mathematical rules. As one of the tools of mathematics, linear systems have multiple uses in the real world. Life is full of situations when the output of a system doubles if the input doubles, and the output cuts in half if the input does the same. That’s what a linear system is, and any linear system can be described with a linear equation.

## In the Kitchen

If you've ever doubled a favorite recipe, you’ve applied a linear equation. If one cake equals 1/2 cup of butter, 2 cups of flour, 3/4 tsp. of baking powder, three eggs and 1 cup of sugar and milk, then two cakes equal 1 cup of butter, 4 cups of flour, 1 1/2 tsp. of baking powder, six eggs and 2 cups of sugar and milk. To get twice the output, you put in twice the input. You might not have known you were using a linear equation, but that’s exactly what you did.

## Melting Snow

Suppose a water district wants to know how much snowmelt runoff it can expect this year. The melt comes from a big valley, and every year the district measures the snowpack and the water supply. It gets 60 acre-feet from every 6 inches of snowpack. This year surveyors measure 6 feet and 4 inches of snow. The district put that in the linear expression (60 acre-feet/6 inches) * 76 inches. Water officials can expect 760 acre-feet of snowmelt from the water.

## Just for Fun

It’s springtime and Irene wants to fill her swimming pool. She doesn’t want to stand there all day, but she doesn’t want to waste water over the edge of the pool, either. She sees that it takes 25 minutes to raise the pool level by 4 inches. She needs to fill the pool to a depth of 4 feet; she has 44 more inches to go. She figures out her linear equation: 44 inches * (25 minutes/4 inches) is 275 minutes, so she knows she has four hours and 35 minutes more to wait.

## Looking Good

Ralph has also noticed that it’s springtime. The grass has been growing. It grew 2 inches in two weeks. He doesn’t like the grass to be taller than 2 1/2 inches, but he doesn’t like to cut it shorter than 1 3/4 inches. How often does he need to cut the lawn? He just puts that calculation in his linear expression, where (14 days/2 inches) * 3/4 inch tells hims he needs to cut his lawn every 5 1/4 days. He just ignores the 1/4 and figures he’ll cut the lawn every five days.

## Everywhere

It’s not hard to see other similar situations. If you want to buy beer for the big party and you’ve got $60 in your pocket, a linear equation tells you how much you can afford. Whether you need to bring in enough wood for the fire to burn overnight, calculate your paycheck, figure out how much paint you need to redo the upstairs bedrooms or buy enough gas to make it to and from your Aunt Sylvia’s, linear equations provide the answers. Linear systems are, literally, everywhere.

## Where They Aren’t

One of the paradoxes is that just about every linear system is also a nonlinear system. Thinking you can make one giant cake by quadrupling a recipe will probably not work. If there’s a really heavy snowfall year and snow gets pushed up against the walls of the valley, the water company’s estimate of available water will be off. After the pool is full and starts washing over the edge, the water won’t get any deeper. So most linear systems have a “linear regime” — a region over which the linear rules apply— and a “nonlinear regime” — where they don’t. As long as you’re in the linear regime, the linear equations hold true.