Simultaneous equations are a system of equations that are all true together. You must find an answer or answers that work for all the equations at the same time. For example, if you’re working with two simultaneous equations, even though there may be a solution that makes one of the equations true, you must find the solution that makes both equations true. Simultaneous equations can be used to solve everyday problems, especially those that are more difficult to think through without writing anything down.

## Rate, Distance and Time

You can calculate the best routes for your running or cycling schedule by creating a mathematical expression that takes into account the distance and your average speed for various parts of the route. You can use the equations to set different goals, such as to maximize time for build endurance, or to maximize speed for performance.

## Planes, Trains and Automobiles

The same formula used to calculate running times can be used to determine speed, distances and time duration when traveling by car, plane or train and you want to know the values for the unknown variables in your travel situations.

## The Best Deal

You want to find out the better deal when renting a car, and you're comparing two rental companies. By putting the variable and fixed costs, such as the per-mile and daily rate, into an algebraic expression, then solving for the total cost, you can see which company saves you money for different amounts of driving.

## The Best Plan

You can use this same process with a system of equations when trying to decide on the best cell phone plan, determining at how many minutes both companies charge the same amount and deciding from there which is the best plan for you and your intended usage.

## Deciding on a Loan

Simultaneous equations can be used to determine the best loan choice to make when buying a car or a house when you consider the duration of the loan, the interest rate and the monthly payment of the loan. Other variables may be involved as well. With the information at hand, you can calculate which loan is the best choice for you.

## Cost and Demand

Simultaneous equations can be used when considering the relationship between the price of a commodity and the quantities of the commodity people want to buy at a certain price. An equation can be written that describes the relationship between quantity, price and other variables, such as income. These relationship equations can be solved simultaneously to determine the best way to price the commodity and sell it.

## In the Air

An air traffic controller can use simultaneous equations to ensure two airplanes don’t intersect at the same time.

## The Best Job for the Money

Systems of equations can be used when trying to determine if you’ll make more money at one job or another, taking multiple variables into account, such as salary, benefits and commissions.

## Investing Wisely

You can use simultaneous equations to decide on your best investment option, taking into account the duration of the investment, the interest it will accrue, as well as other variables that will affect the end result. If you know the amount you’d like to accrue, you can set the options equal to each other and figure out which option is best for your situation.

## Mixing It Up

With respect to mixtures, simultaneous equations can be used for achieving a certain consistency in a resultant product, which is dependent on the consistency of the compounds mixed together to produce it.

#### References

- "Dr. Math Explains Algebra"; The Math Forum; 2004
- "Algebra Survival Guide"; Josh Rappaport; 2002
- Math Warehouse: Systems of Linear Equations;
- The Math Page: Word Problems That Lead to Simultaneous Equations; Lawrence Spector; 2011