Frequently Asked Questions

Simultaneous equations are a set or group of two or more equations, with each one having one or more "unknowns" in common (or "variables" in the language of mathematics). Each equation will either specify a numeric value for one (or possibly more than one) unknown, or it will define how one or more unknowns can be expressed as a combination of one or more other unknowns. The equations are called "simultaneous" because they must be studied and solved together in their entirety to figure out the answer to each individual unknown.

If you click on the "How To Solve" tab, you’ll see a description of several techniques that you can use to solve the simultaneous equation puzzles on SimEqPuzzles.com. Additionally, searching the Internet for the keywords "simultaneous equations" will return a number of websites that give examples of simultaneous equations, with tips and tricks for solving them.

It is sometimes possible for a group of simultaneous equations to have more than one solution (i.e., some of the unknowns may have more than one possible numeric value). This generally happens when there is some form of duplication somewhere among the equations. A duplication occurs when one of the equations is a mathematical combination of (or can be derived from) one or more other equations. This makes the duplicate equation unnecessary, and usually leads to some unknowns having more than one possible answer. On SimEqPuzzles.com, however, each puzzle has one and only one solution (i.e., each unknown can have just one possible numeric answer).

"I figured it out a different way" likely means that you figured out the answers to a puzzle’s unknowns in a different order than the solution presented, or you found a way to combine equations in a way that’s different from the solution revealed. This may occur if you happen to notice equations that could be combined or simplified in a different order than the solution given. In a set of simultaneous equations, it is very common to have smaller subsets of equations that can be simplified independently and in any order. Depending on which of these subsets you happen to notice first, you may combine equations and solve the unknowns in a different order or different way than the solution given. This is normal, and perfectly OK.

Adjoined letters (i.e., a string of two or more letters without a math symbol between them) represent a number with more than one digit. In the example above, AB and CD are 2-digit numbers, and DEF is a 3-digit number. In the 2-digit numbers AB and CD, A and C stand for digits in the ten’s position, while B and D denote digits in the unit’s position. Stated mathematically, AB = 10×A + B and CD = 10×C + D. In the 3-digit number DEF, D stands for a digit in the hundred’s position, E for a digit in the ten’s position, and F for a digit in the unit’s position. Mathematically, this means that DEF = 100×D + 10×E + F. (While mathematicians frequently can and do adjoin two or more letters as a short-cut for multiplication, the puzzles on SimEqPuzzles.com will not do this.)

The ★s next to the puzzle’s date represent the puzzle’s "degree-of-difficulty". Puzzles can be rated with 1, 2, 3, 4, or 5 ★s. 1★ is the least difficult, 5★s is the most difficult. Puzzles usually become harder as the week progresses, beginning with the easiest on Monday, and ending with the hardest on Sunday.

Each day’s puzzle will become available for you to solve between midnight and noon Central Time (USA). On holidays, there occasionally may be a delay. Most new puzzles become available between midnight and 2 am Central Time (USA).

Here's an example of using simultaneous equations to help you decide between two competing financial alternatives, with each having both fixed and variable costs. Let's say you've narrowed your choice of cellphone plans down to 2 providers -- BigBadBandWidth and MultiMegaMinutes. Both plans offer the same coverages, similar call quality, and comparable features, so the only appreciable difference between them is their cost. The best plan from MultiMegaMinutes costs $29.50 per month plus 6¢ per calling minute. BigBadBandWidth's best plan costs only $18.50 per month, but costs an additional 11¢ per calling minute. You want to choose the plan that costs you the least.

The total cost of both MultiMegaMinutes' and BigBadBandWidth's cellphone plans can be written as algebra equations:

   eq.1)  29.50 + 0.06 × A = B    (MultiMegaMinutes)
   eq.2)  18.50 + 0.11 × A = B    (BigBadBandWidth)

where:

   29.50 = the fixed monthly cost of MultiMegaMinutes' plan (in $)
   18.50 = the fixed monthly cost of BigBadBandWidth's plan (in $)
   0.06 = the calling cost of MultiMegaMinutes' plan (in $ per minute)
   0.11 = the calling cost of BigBadBandWidth's plan (in $ per minute)
   A = the number of monthly calling minutes where both plans cost the same
   B = the total monthly cost (in $) where both plans cost the same

Solving these two equations will find the numbers for A and B where the monthly cost of both plans is identical.
First, substitute 29.50 + 0.06 × A for B from eq.1 into eq.2:

   18.50 + 0.11 × A = 29.50 + 0.06 × A

Next, subtract 18.50 from both sides of this equation:

   18.50 + 0.11 × A – 18.50 = 29.50 + 0.06 × A – 18.50   which becomes
   0.11 × A = 11.00 + 0.06 × A

Now, subtract 0.06 × A from each side:

   0.11 × A - 0.06 × A = 11.00 + 0.06 × A - 0.06 × A   which becomes
   0.05 × A = 11.00

Next, divide both sides by 0.05:

   0.05 × A ÷ 0.05 = 11.00 ÷ 0.05   which makes
   A = 220

Finally, substitute 220 for A in both eq.1 and eq.2:

   eq.1)  29.50 + 0.06 × 220 = 29.50 + 13.20 = 42.70 = B    (MultiMegaMinutes)
   eq.2)  18.50 + 0.11 × 220 = 18.50 + 24.20 = 42.70 = B    (BigBadBandWidth)

So, you've found that both plans would cost the same ($42.70) at 220 minutes per month. Each additional minute above 220 will cost you 11¢ on BigBadBandWidth's plan, but only 6¢ on MultiMegaMinutes' plan. So you'd buy the MultiMegaMinutes plan if you expect to use your cellphone more than 220 minutes per month. Conversely, each minute less than 220 will save you 11¢ with BigBadBandWidth, but only 6¢ with MultiMegaMinutes. So you'd go with BigBadBandWidth if you use your cellphone less than 220 minutes per month.

All of SimEqPuzzles.com's features work well with the web-browsers Chrome, Firefox, and Microsoft Edge. Not all of SimEqPuzzles.com's features work as well in Internet Explorer (IE). In particular, the Puzzles page's Scratchpad is not as well-behaved in IE as in the other browsers. We suggest using a browser other than IE (such as Firefox, Chrome, or Edge).

Some puzzles contain special math characters in their hints and solutions. These characters may not display properly on cellphones or tablets using the default Android browser. We suggest using a different browser (such as Firefox) instead.

If you are new to solving simultaneous equation problems, or if it has been a while since you've solved one, the puzzles on SimEqPuzzles.com could look a little intimidating.  Most of the puzzles have only one number or constant in them (some puzzles don't have any).  So a puzzle with lots of equations, lots of unknowns, and only one number may seem difficult (if not impossible) to solve.

So if simultaneous equation problems are new to you, or if you haven't solved one in a long time, first try reviewing the examples and the explanations on the "How To Solve" page.  This will provide you with a good foundation and basic understanding of how to solve simultaneous equations.  Then try solving some of the easier 1-star problems in the "Puzzles" page's archives.  After you've become comfortable solving 1-star puzzles, try a few of the slightly harder 2-star puzzles next.  By gradually progressing from 1-star to 2-star puzzles, and then moving on to 3- and 4- and 5-star puzzles, you will become more comfortable with and proficient at solving simultaneous equation problems in general.  Soon you will be able to solve any puzzle with any degree of difficulty on SimEqPuzzles.com.