Solving Systems of Equations Explained!

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Solving Systems of Equations Explained!

This free step-by-step guide will teach you everything you need to know about solving systems of equations in math.

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Solving Systems of Equations: Everything You Need to Know

Solving systems of equations can seem intimidating, especially when you see more than one equation shown on a graph. However, if you know how to graph a function on the coordinate plane or on a graphing calculator, then you can become a master of solving systems of equations.

This guide also includes a very handy system of equations solver that you can use to check your work and graph linear systems on your computer.

But first…

If you need some refreshers on the foundational skills required to understand how to solve systems of equations, you may find these free algebra resources to be helpful:

You’ll also notice that the graphics in this lesson rely on using different colors to differentiate between different functions (this strategy is very helpful for keeping your thoughts organized and preventing confusion). If you are following along using graph paper, then it is highly recommended that you use colored highlighters or markers. However, this is only a suggestion, and you can still learn to solve systems of equations using a pen or pencil.

 Are you ready to get started?

System of Equations Definition

A system of equations is when there are two or more equations that share the same variables.

 For example, here is a system of equations for two linear functions:

y = x + 1 & y=-2x + 1

 Notice that both of these equations are shown on the graph in Figure 1. 

(Again, if you need a refresher on how to graph lines in y=mx+b form, watch this quick video tutorial)

Figure 1 (graph courtesy of desmos.com/calculator)

Figure 1 (graph courtesy of desmos.com/calculator)

 

The solution to a system of equations is the point (or points) where the lines intersect.

 So, in this example, the solution to the system of equations is the point (0,1), since this is where the two lines intersect. 

Figure 2: What do lightsaber fights and linear systems have in common? (Image: Mashup Math MJ)

Types of Solutions to Systems of Equations

Does every system of equations have a solution?

There are actually three kinds of solutions to a system of equations:

  • One Solution (as seen in Figure 1)

  • No Solution

  • Infinitely Many Solutions

 
Figure 3

Figure 3

 

Key Takeaways:

One Solution: The systems intersect at only one point.

No Solution: The lines are parallel and do (and never will) intersect

Infinitely Many Solutions: Two or more identical and overlapping graphs that intersect everywhere!

Confused? That’s ok. Just keep the three solution types in mind as we work through an example of each type of solution that will help you to better understand how to solve systems of linear equations.

Systems of Linear Equations Examples

Example 01: One Solution

Find the solution to the following system of equations:

Figure 4. Notice that the answer to this example is a decimal, which is totally fine.
 

The first step to finding the solution to this system of equations is to graph both lines as follows:

Snip20210114_13.png
 

Notice that the ONLY intersection point for this system of equations is at (2,5).

Snip20210114_14.png
 

Remember that (2,5) is an (x,y) coordinate where x=2 and y=5. To confirm that you answer is correct, you can substitute x=2 and y=5 into both equations to see if your answer checks out as follows.

Snip20210114_16.png
 

Note that, since this system of equations has only one solution, (2,5) is the only point that will work. You can try substituting x and y values for any other coordinate and you will never find another one that works out.

Final Answer: The solution is (2,5)


Example 02: No Solution

Find the solution to the following system of equations:

Snip20210114_20.png
 

Just like the last example, graph both equations on the coordinate plane as follows:

Snip20210114_21.png
 

Notice that both equations have the same slope (+5/4). Since parallel lines have the same slope, it makes sense that the lines are the graph are parallel to each other. And, since parallel lines never intersect, these two lines will never intersect, and therefore there is no solution to this system of equations.

Final Answer: No Solution (because the lines are parallel)


Example 03: Infinitely Many Solutions

Find the solution to the following system of equations:

Snip20210114_26.png
 

Notice that the second equation is not in y=mx+b form, so you will have to rearrange it to isolate the y before you can graph:

Isolate the y by subtracting 3 from both sides.

Isolate the y by subtracting 3 from both sides.

What do you notice?

What do you notice?

Now we have two equations in our system: y=-4x-3 and… y=-4x-3. After rearranging the second equation, we can see that both equations are identical.

What does this mean for the graph and the solution to our linear system? Let’s find out by graphing the first equation (Figure 5) and then the second equation (Figure 6)

Figure 5

Figure 5

Figure 6

Figure 6

Since the equations are identical, the lines are graphed right on top of each other and they intersect everywhere at every point that both lines pass through.

Therefore, every point on the line is a solution—and since lines have an infinite number of points, this system has an infinite number of solutions.

Final Answer: Infinitely Many Solutions


Systems of Equations Solver

When you initially learn how to solve systems of equations, we recommend using graph paper and a straight-edge to graph your equations and find your solution (and use substitution to check your work as we did in example 01).

However, after you become more comfortable working with systems of equations, you may benefit from using systems of equations solver tool like a graphing calculator to graph lines and find intersection points rapidly.

If you don’t have a graphing calculator, there is an awesome FREE system of equations solver calculator available via www.desmos.com/calculator.

*Note that you have to input the equations in y= form in order for the Desmos systems of equations solver to work.

For example, you can use the Desmos Systems of Equations Solver to find the solution to the system:

y=(3/5)x-8 & y=(-8/5)x+3

1.) Type each equation into the left-hand column

2.) Locate the intersection point (you may need to zoom out)

3.) Click on the intersection point to find the coordinates.

The solution is (5,-5)

Screenshot of Desmos.com/calculator

Screenshot of Desmos.com/calculator


Systems of Equations Video Lesson

If you are a visual learner and would like to review this step-by-step guide to solving systems of equations as a video tutorial, check out these free tutorials:



Keep Learning with These Free Math Guides:


Share your ideas, questions, and comments below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math . You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

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cm to m: 2 Easy Steps

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cm to m: 2 Easy Steps

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Converting CM to M: Everything You Need to Know

Are you looking to learn how to convert centimeters to meters (cm to m)?

Before you learn the absolute easiest way to convert cm into m (with and without a calculator), let’s do a quick review of some very important vocabulary terms (trust me, this will come in handy very soon):

  • Centimeter (cm) is a metric unit of length that equals one-hundredth of a meter.

  •  Meter (m) is the fundamental unit of length in the metric system. Note that a meter is approximately 39.37 inches or 3 feet and 3 inches in the imperial system)

  • Note that the abbreviations for centimeters (cm) and meters (m) are meant to be expressed as lower-case letters.

The key takeaway from these definitions is that a meter is the standard unit of length in the metric system and that a centimeter is equal to one-hundredth of one meter (which means that there are one hundred centimeters in one meter). Centimeters are very small compared to meters, and:

  • One hundred centimeters make up one meter.

  • One meter is made up of one hundred centimeters.

Are you starting to see a relationship between centimeters and meters?

 

Figure 1: Centimeters Compared to Inches (image via Mashup Math FP)

 
  • Remember that meters and centimeters are metric system units of measurement (not imperial).

  • For reference, some things that measure approximately 1 meter are the width of a large refrigerator, the depth of the shallow end of a swimming pool, and the height of a typical kitchen countertop.

 

Figure 2: cm to m Relationship (image via Mashup Math FP)

 

If you can understand the relationship between cm and m (and m and cm) displayed in Figure 2, then you will be able to convert cm into m using the easy two-step method shown below!

Converting cm to m Examples

You can convert centimeters to meters by following these two easy steps.

(Note: First, you will learn how to convert cm into m by hand and then later on using a calculator).

cm to m Example 01: Convert 500 cm to meters

Let’s start with a relatively simple example, where you have to convert 500 centimeters into meters.

Step One: Divide the Number of Centimeters by 100, as follows:

500 ÷ 100 = 5

Step Two: Change the Units to Meters

500 cm = 5 m

That’s it! (see Figure 3 for a more detailed explanation)

Final Answer: 500 centimeters equals 5 meters

Figure 3

Figure 3

 

By following the two simple steps above, you can always convert cm to m even if you don’t have a calculator. Now, let’s take a look at another example.

Here is another example of how to convert cm to m:

cm to m Example 02: Convert 886 centimeters into meters

In this example, you can convert cm to m using the same two-step method used in the previous example:

Step One: Divide the Number of Centimeters by 100, as follows:

886 ÷ 100 = 8.86

Step Two: Change the Units to Meters

886 cm = 8.86 m

Final Answer: 886 centimeters equals 8.86 meters

Figure 4. Notice that the answer to this example is a decimal, which is totally fine.

Figure 4. Notice that the answer to this example is a decimal, which is totally fine.

 

What if you flip cm to m?

Before you move onto learning about the cm to m calculator and trying some free practice problems, let’s take a super quick look at the relationship between the reverse of cm to m: converting meters into centimeters.

As you may have predicted, the conversions between cm and m are reversible. When going form cm to m, you divide by 100. When going in reverse, from m to cm, you multiply by 100.

Fun Fact: If you start with meters and convert to centimeters and then convert back to meters, or vice versa, you will end up with the same number that you started with (some students like to perform these calculations to check that their work is correct).

Example 03: Convert 8.86 meters into centimeters

Notice that this example is the reverse of Example 02 and you have to convert to m into cm. If the fun fact is true, then the final answer has to be 886 centimeters. Let’s see if it works out:

Step One: Multiply the Number of Meters by 100, as follows:

8.86 x 100 = 886

Step Two: Change the Units to centimeters

886 cm = 8.86 m

Final Answer: 8.86 meters equals 886 centimeters

Pretty cool, right?

Figure 5

Figure 5

 

cm to m Calculator

Via calculatorsoup.com

Screenshot from calculatorsoup.com

If you need a fast and easy way to convert between different units of measurement, including centimeters to meters (cm to m), then you can take advantage of the many free online centimeters to meters conversion calculators that are available.

This free distance conversion calculator from www.calculatorsoup.com will quickly make conversions from cm into m and vice versa, but it will not show you any of the work or the previously mentioned two-step process. While this cm to m calculator is a handy tool, it will not help you understand the process behind making distance conversions or the relationship between centimeters and meters.

To use the cm to m calculator, simply input the number of centimeters in the Value to Convert box and choose to convert from centimeters to meters. If you do not input the numbers correctly, you will not get a correct cm to m conversion.


Converting cm to m Practice Problems

Looking for some extra practice converting cm into m (and m into cm)?

The following centimeters to meters practice problems will give you plenty of opportunities to apply the two-step process to converting cm to m or to use a cm to m calculator to make conversions. There is also an answer key at the bottom of this post.

1.) Convert 1,000 cm to m

2.) Convert 2,100 cm to m

3.) Convert 55 m to cm

4.) Convert 3.75 m to cm

5.) Convert 10,222 cm to m

6.) Convert 0.5 m to cm

7.) Convert 1 Million cm to m

8.) Convert 0.01 m to cm

9.) Convert 774 cm to m

10.) Convert 6 Billion cm to m

And if you are looking for some real-world practice problems involving measurement conversions, including converting cm to m, check out this free measurement conversion video lesson:

 
 

Answer Key:

1.) 10m

2.) 21 m

3.) 5,500 cm

4.) 375 cm

5.) 102.22 m

6.) 50 cm

7.) 10,000 m

8.) 1 m

9.) 7.74 m

10.) 10 million m



Share your ideas, questions, and comments below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

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By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math . You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

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10 Free Maths Puzzles with Answers!

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10 Free Maths Puzzles with Answers!

10 Fun (and Free) Maths Puzzles with Answers

A Post By: Anthony Persico

Are You Ready for These Super Fun (and Slightly Brain-Bending) Maths Puzzles?

Every month, thousands of individuals, young and old, search the internet for maths puzzles with answers. Why? Because working on challenging and engaging maths puzzles is more than just a recreational activity. In fact, studies show that working on maths puzzles has several educational benefits including boosting interest in mathematics, improving problem-solving skills and algebra skills, and cultivating reflective learning abilities.

So, if you’re one of these individuals looking to reap the benefits of working on challenging maths puzzles (while having some serious fun while doing so), then you’re in the right place!

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Now that you know the benefits of challenging your mind by trying to solve maths puzzles, you’re just about ready to try and solve today’s collection of super fun maths riddles.

Note that these maths riddles with answers are appropriate for individuals ages 12 and up.

Before you get started, here are a few tips to keep in mind when attempting to solve any of today’s maths puzzles:

  • Read every maths puzzle carefully and allow yourself to think for a bit before getting started.

  • Use proven strategies like visualizing, diagram drawing, and trial-and-error.

  • You’re going to struggle! When you find yourself feeling discourages and/or messing up, that means you are in the process of reaping the benefits of solving maths puzzles and improving your problem-solving skills!

  • Whenever you find an answer, ask the question “does this solution make sense?”

  • If at any time you get stuck, close the page and take a break. This is a good time to take your mind off of the problem by doing something else like going out for a walk. The puzzle will likely become more manageable when you revisit it later on.

Practice Problem: How Many Squares?

Before you attempt the 10 Maths Puzzles with Answers on your own, you may want to get warmed-up with a relatively easy maths puzzle just to get your brain fired up and ready for some more challenging problems later on.

Of course, you can choose to continue on and skip this practice problem if you’d like.

 
 

Here is the problem:

How many SQUARES are in the 3x3 grid?

How to solve:

This is a relatively straightforward maths puzzle to solve, provided that you consider two pieces of information:

  • A square is a plane figure with four equal sides and four right angles.

  • Some squares are overlapping in the diagram.

Want to try the problem on your own? If so, don’t go any further. Stop here and try to solve the puzzle and come on back when you’re finished.

The most common way to solve this problem is to consider all the squares, from smallest to largest, and count them as follows:

 

Final Answer: 14 Total Squares

Wasn’t that fun? Now you are ready to move onto some more challenging maths puzzles!


10 Fun Maths Puzzles with Answers

Each of the following math puzzles with answers includes an image graphic. Click on any image to enlarge. The complete answer key for all 10 maths puzzles is included at the bottom of the page.

Have fun!

1.) Maths Puzzles 01 of 10: How Many Rectangles?

This problem may seem very similar to the practice problem, but the subtle difference between Squares and Rectangles is a very big deal that makes this problem rather tricky. (Click here to learn more about this controversial maths puzzle)

Can you solve it?

(Hint: By definition, is a square a rectangle?)


2.) Maths Puzzle 02 of 10: Factors and Fruit

Problem: If each piece of fruit in the diagram below is equal to one of the following numbers: 1, 2, 3, or 5, then what is the value of each fruit so that both equations are true?


3.) Maths Puzzle 03 of 10: Moon Math

Problem: What is the value of the missing number “?” in the lunar diagram below?

 

Hint: Look for a pattern.


4.) Maths Puzzle 04 of 10: Which is More Pizza?

Problem: Which deal gets you the most pizza (if each deal costs the same amount):

  • One 18” pizza pie, or

  • Two 12” pizza pies

Can you solve it?

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Use the area of a circle formula: A= π(r^2)


5.) Maths Puzzle 05 of 10: Super Maths!

Problem: Find the value of each symbol and the ‘?’ in the puzzles below:

Puzzle 1

Puzzle 2

Puzzle 3


Are you looking for more super fun Maths Riddles, Puzzles, and Brain Teasers?

The best-selling workbook 101 Math Riddles, Puzzles, and Brain Teasers for Ages 10+! is now available as a PDF download. You can get yours today by clicking here.


6.) Maths Puzzle 06 of 10: How Many Triangles

Problem: What is the total number of triangles in the diagram below?


7.) Maths Puzzle 07 of 10: Matchsticks Maths

Problem: Make the maths equation true by moving ONE AND ONLY ONE matchstick?

(*Bonus if you can find all three possible answers)

See Also: video tutorial on solving the matchstick maths problem.

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8.) Maths Puzzle 08 of 10: How Many Total Handshakes?

Problem: If there are 20 people in a room and they shake every other person’s hand once and only once, how many handshakes would take place?

Handshakes.jpg

9.) Maths Puzzle 09 of 10: Shopping Spree!

Problem: At the mall, the total cost of a pair of shoes and a hoodie is $150. The cost of the hoodie is $100 more than the cost of the pair of shoes. How much does each item cost?


10.) Maths Puzzle 10 of 10: Parking Lot Puzzle

Problem: In the diagram below, what is the number of the parking spot occupied by the car?


11.) Bonus Maths Puzzle! : The Big Triple!


MATHS PUZZLES ANSWER KEY:

  1. 36 Total Rectangles (click here to learn more about how to solve this problem)

  2. Melon=5, Apple=2, Grapes=3, Lemon=1

  3. ?=8 (the relationship between the diagonals are cubes and cube roots)

  4. Area of 18” Pizza: ~254.3 square inches > Area of two 12” pizzas: 226 square inches

    (click here to learn more about how to solve this problem)

  5. Puzzle #1: DK=12, Bananas=8, Barrell=4, ?=24

    Puzzle #2: Mario=12, Peach=12, Toad=6, Bowser=0, ?=30

    Puzzle #3: Van=8, Zombie=9, Scooby=5, ?=77

  6. 27 Total Triangles

  7. 8-4=4, 5+4=9, 0+4=4 (Here is a great video explanation)

  8. 190 handshakes (19+18+17+16+...+3+2+1=190)

  9. Hoodie costs $125, Shoes cost $25

  10. 87 (click here to learn how to solve this problem)

11. (BONUS!) 1+2+3=6 and 1 x 2 x 3 = 6

Click here to sign up for our math education mailing list to start getting free K-12 math activities, puzzles, and lesson plans in your inbox every week!


Do YOU Want More Fun Math Riddles, Puzzles, and Brain Teasers?

Check out our math riddle videos on YouTube!


Did I miss your favorite math riddle for adults? Share your thoughts, questions, and suggestions in the comments section below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math and an advisor to Amazon Education's 'With Math I Can' Campaign. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

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We Spent All Morning Working on This Rectangle Math Puzzle. Can You Solve It?

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We Spent All Morning Working on This Rectangle Math Puzzle. Can You Solve It?

The controversial real answer is finally revealed. Can you solve it?

A Post By Anthony Persico

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It’s no secret that we are obsessed with math riddles, puzzles, and brain teasers here at Mashup Math. Why? Because there are not many better feelings in life than finally reaching an aha moment after mulling over a brain-bending math brain teaser for a while (sometimes hours!) and figuring out the correct answer.

Like most math fans, we love problem-solving, and, like most math fans, we can be rather hardheaded when it comes to believing that our answer is correct and that any other answer is obviously wrong.

So, it was no surprise that a seemingly simple yet controversial math puzzle forced us to drop everything we were doing and spend an entire morning arguing about who had found the elusive right answer.

We have been sharing the rectangle math riddle with math fans for a while—well, sort of. The original math puzzle was nearly identical, except that it asks How Many Squares? (Not How Many Rectangles). This problem, shown below, is not too difficult to solve and it is appropriate for students at the elementary levels and above.

The Original Problem: How Many Squares?

When squares are involved, most people can figure out that the answer is 14 total squares.

(Hint: if you are unsure of how to solve the Squares Problem, remember that some of the squares are overlapping).

 

14 Total Squares

 
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But What If I asked How Many Rectangles?

The How Many Squares Problem? is relatively simple and easy to solve. So, why does replacing Squares with Rectangles cause so much controversy?

This craziness all started when I received a text from a friend asking me if I had a quick math puzzle that I could send her to share with her son, a 4th-grade student, to try and solve before bed. With the How Many Squares? puzzle in mind, I quickly sketched the puzzle on a piece of lined paper using a Sharpie, took a picture with my phone, and sent it over to her. And, since I already had the picture, I also tweeted it out to all of our followers and went on with my evening.

This was the Tweet that launched a thousand different answers:

It wasn’t until the next morning that I saw that the tweet had over 200 replies and that I had accidentally asked How Many Rectangles? instead of How Many Squares?

And so, the controversy began.

Dozens of different solutions were being shared and several arguments breaking out.

I could have spent my entire day reading Twitter replies, but instead, I shared the puzzle with some friends who happen to be self-proclaimed Geometry wizards to see if there was a consensus solution. And I myself was on a quest to figure out why such a seemingly simple math puzzle had caused so much controversy.

Want to try the problem on your own? If so, don’t go any further. Stop here and try to solve the puzzle and come on back when you’re finished.

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Gee, that was pretty quick! Are you ready to see the solution?

Unlike many popular math riddles and brain teasers that are purposely ambiguous and can have multiple answers, this math puzzle has one single, undeniable answer, and it’s 36 total rectangles.

Before we look at why 36 is the solution, let’s take a look at some of the most common responses on Twitter:

As you can see, there is a wide variety of answers and only a small percentage of people found the correct answer.

Some people even went as far as to claim that there are actually zero rectangles and blamed my inability to draw straight lines freehand…

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I'll be sure to use a ruler next time.

Anyways, here is one more notable Tweet that will launch our discussion into why the actual answer is 36 rectangles:

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How to Solve the Rectangles Math Riddle

Miri’s tweet is notable because she draws attention to a very important math fact that many people are forgetting or unaware of—that a square is a rectangle.

What!?

It’s true. By definition, a rectangle is a plane figure with four straight sides and four right angles, especially one with unequal adjacent sides.

And a square certainly fits this definition. In fact, a square a special kind of rectangle (check out the Is a Square a Rectangle? video below for a more in-depth explanation.

 
 

Now that we know that a square is a rectangle, we can use the same approach to How Many Squares? problem to solve the Rectangle Math Puzzle.

Here’s a handy diagram of how to find each possible rectangle:

Diagram.jpg

Rectangle Math Puzzle Solution Breakdown

Start by counting all of the squares, in which there are 14 in total (shaded in pink). Did you remember to count the large perimeter square?

Squares: 14

Then move onto counting ALL of the rectangles:

1x3 Rectangles (Blue): 6

2x3 Rectangles (Green): 4

1x2 Rectangles (Orange): 12

Finally, find the sum of all of the rectangles:

14 + 6 + 4 + 12 = 36 Total Rectangles


Do YOU Want More Fun Math Riddles, Puzzles, and Brain Teasers?

Wasn’t that fun? If you want to take on more brain-bending math puzzles to sharpen your mind and improve your problem-solving skills, check out the links below:

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The PEMDAS Rule Explained! (Examples Included)

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The PEMDAS Rule Explained! (Examples Included)

The PEMDAS Rule Explained!

What is the PEMDAS Rule and how does it apply to the math order of operations?

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A Post By Anthony Persico

What is the PEMDAS rule for math order of operations and solving problems? When studying math, you learn about a process called the order of operations. This process is a rule that must be followed when solving math problems that have multiple operations such as subtraction, addition, multiplication, division, groupings, and/or exponents.

There are many memory tricks for remembering the math order of operations in the correct order, but the most popular is the PEMDAS rule.

The PEMDAS Rule is a mnemonic that stands for:

P: Parenthesis

E: Exponents

M: Multiplying

D: Dividing

A: Adding

S=Subtracting

The operations included in the PEMDAS rule are performed left to right.

Additionally, the PEMDAS rule for recalling the math order of operations has a few important sub-rules that must also be followed if you want to use PEMDAS correctly (and get the correct answers to math problems). These important sub-rules relate to the relationships between multiplying/dividing and adding/subtracting.

These important PEMDAS rule sub-rules are explained in detail in the next section:

PEMDAS Rule: Key Points

The PEMDAS rule has been around for decades as a tool for helping students to remember the math order of operations. Many prefer to simply remember the mnemonic PEMDAS (pronounced PEM-DAHS), while others choose to remember the phrase Please Excuse My Dear Aunt Sally.

However, you choose to remember the PEMDAS rule is not as important as remembering the previously mentioned sub-rules? Why are the sub-rules to the PEMDAS rule so important? Because the sub-rules often make the difference between getting a correct or incorrect answer to a math problem.

The PEMDAS rule may not be perfect, but if you can remember the sub-rules, it can be a useful tool for helping you to correctly apply the math order of operations and getting correct answers on both simple and complex math problems provided that you know the important sub-rules.

Important Sub-Rules to the PEMDAS Rule:

1.) P: Perform operations inside of parenthesis or groups before you do anything else (if there are no groups or parentheses, you can skip this step).

2.) E: Next, after performing operations inside of parenthesis and groupings (if there are any), apply any exponents (if there are no exponents, you can skip this step).

3.) M/D: Next, after the parentheses and groups and the exponents, perform multiplying/dividing from left to right based on whichever operation is first).

★ Just because M comes before D in the PEMDAS rule doesn’t mean that you will always perform multiplication before division.

4.) A/S: Finally, after multiplying and/or dividing, perform adding/subtracting from left to right based on whichever operation is first).

★ Just because A comes before S in the PEMDAS rule doesn’t mean that you will always perform addition before subtraction

= Extremely Important

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PEMDAS Rule Examples

Now that you know what the PEMDAS rule for remembering the math order of operations stands for, it’s time to learn how to use the rule to solve math problems and get correct answers.

Why? Because there is zero educational value in remembering what the PEMDAS rule stands for if you have no clue how to apply it to math order of operations.

Remember that the PEMDAS rule is only useful if you also remember the key sub-rules that were displayed in the last section.

With the sub-rules in mind, the next section will work through several examples of how to correctly apply the PEMDAS rule when it comes to math order of operations and problem-solving.

PEMDAS Rule Ex. 1: (3+1) x 4

First, solve whatever is in the groupings (parentheses):

3+1=4

Next, multiply: 4 x 4 = 16

Final Answer: 16

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PEMDAS Rule Ex. 2: 27 ÷ (8-5)^2

 Again, perform operation inside of the groupings first.

Inside parentheses: 8-5 = 3.

The next step to evaluate the exponents: 3^2=9

The last step is to divide: 27÷9 = 3

Final Answer: 3

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PPEMDAS Rule Ex. 3: 10 x 6 + 1

Notice that this example does not include groupings or any exponents. Therefore, you can skip the P and E in the PEMDAS rule and start with M/D.

Since multiplying/dividing comes before adding/subtracting, you can solve this problem by moving from left to right as follows:

 10x6 = 60

60 + 1 = 61

Final Answer: 61

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PEMDAS Rule Ex. 4: 75 - 10 x 5

According to the PEMDAS rule, multiplying/dividing comes before adding/subtracting so you can NOT solve this problem by moving from left to right.

The PEMDAS rule requires you to multiply first and then perform subtraction second as follows:

10 x 5 = 50

75 – 50 = 25

Final Answer: 25

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PEMDAS Rule Ex. 5: 8 x 8 ÷ 16

Are you ready to apply the important sub-rules?

Notice that there are only two operations in this math example: multiplication and division.

Before we move forward, let’s revisit sub-rule #3:

3.) M/D: Next, after the parentheses and groups and the exponents, perform multiplying/dividing from left to right based on whichever operation is first).

Just because M comes before D in the PEMDAS rule doesn’t mean that you will always perform multiplication before division.

In this problem, you can solve by performing multiplication fist (the left-most operation) and then division second as follows:

8 x 8 = 64.

64 ÷ 16 = 4

Final Answer: 4

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Note: If you followed the PEMDAS rule strictly and solved from left to right, you could still have ended up with 4 as the correct answer. However, this will not always be the case as we will in the next example.

PEMDAS Rule Ex. 6: 42 ÷ 7 x 3

Let’s start by saying that many people will get this simple problem wrong because they forget the key sub-rules to the PEMDAS rule. They will make the mistake of strictly following the PEMDAS rule and performing multiplication before division (since M comes before D in PEMDAS).

Don’t make this mistake!

Remember that ★ Just because M comes before D in the PEMDAS rule doesn’t mean that you will always perform multiplication before division.

In this case, the only operations are multiplying and dividing. This time, division comes first, which is ok. You still solve the problem by moving from left to right as follows:

42÷7=6

6x3=18

Final Answer: 18

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Why is 2 not the final answer? If you failed to apply the PEMDAS rule correctly, you may have made the mistake of performing multiplication before division as follows:

7 x 3 = 21

42 / 21 = 2 (THIS ANSWER IS WRONG!)

(Pro Tip: If this was a multiple-choice question, both 18 and 2 would both be choices. So, be careful!)


PEMDAS Rule: Why Is It Good to Know

It’s impossible to consistently solve math problems (both simple and complex) correctly without understanding how to apply the math order of operations and PEMDAS is an effective tool for recalling them in the right sequence—provided that you also remember the important sub-rules described above.

 The PEMDAS rule and the math order of operations have gained tons of attention in recent years because of viral social media posts sharing seemingly simple math problems that garner thousands of responses (and incorrect answers) due to the fact that many adults can remember “PEMDAS” but not how to apply the actual PEMDAS rule (and corresponding sub-rules).

Since many people can’t figure out the right answer to these simple problems, they are inclined to leave comments and tag friends, which only makes the most more popular on social media.

PEMDAS Rule for Math Order of Operations: Conclusion

The PEMDAS rule is a popular memory tool for recalling the math order of operations. The rule stands for P: Parenthesis, E: Exponents, M: Multiplying, D: Dividing, A: Adding, S=Subtracting.

In general, operations are performed from left to right, but there are very important key sub-rules, namely (1) perform multiplying/dividing from left to right based on whichever operation is first), and perform adding/subtracting from left to right based on whichever operation is first).

Without understanding these sub-rules, the PEMDAS rule becomes extremely unreliable and can lead you to get the wrong answers to simple math problems (see PEMDAS Rule Ex. 6 above).

PEMDAS may not be the best way to remember how to correctly apply the math order of operations, but it can be a reliable tool if and only if you also remember the key sub-rules as well.


The PEMDAS Rule is Good, but the GEMS Rule is Better!

Why is GEMS the Best Way to Teach Order of Operations?


More Free Math Resources for Grades K-8:

 

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