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 know your running pace. You are going to run half of a predetermined route of 14 miles alone and run with a friend for the second half of it. You want to know how long it will take you to run the first half at your pace and the second half at your friend’s pace. Your pace is 7 mph, and hers is 20 percent slower. You can use simultaneous equations to solve this problem. Distance in miles (d) equals the rate in mph (r) multiplied by the time in hours (t). So for this problem, d1 = r1 * t1 and d2 = r2 * t2. You know that d1 = d2, and r2 = 0.8 * r1. So r1 * t1 = 0.8 * r1 * t2, divide by r1 on both sides, and t1 = 0.8 * t2. You know d1 = d2 = 7, so you will run the first 7 miles in 1 hour and you will run the second 7 miles in 1.25 hours or 75 minutes.

### 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. One company charges $30 per day and 40 cents per mile. Another company charges $45 per day and 30 cents a mile. If you can determine when the costs are the same, you can then know which would be the better deal. So you set m = total miles to be driven and c = total cost for each company. Then c = 30 + 0.40 m and c = 45 + 0.30 m. It follows that 30 + 0.40 m = 45 + 0.30m and m = 150. The cost of each company would be the same at 150 miles. Under 150 miles, the first company is cheaper. Above 150 miles, the second company is cheaper.

### 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.