Solving Two‐Step Equations You have probably heard the phrase ‘one‐step equation’ or ‘two‐step equation’ which is a really vague term that refers to basic algebraic equations with one variable. There are endless equations that only take one or two steps to solve but this phrase is a common one in math courses that applies to things like x + 2 = 8 or 10 x − 3 = 9 . The ‘steps’ refer to the number of operations that it takes to solve the problem. Before we go on, let’s review some of the rules about equations and operations. First of all, recall what we talked about in the order of operations tutorial that there are three groups of operations that you can use to combine numbers: 1. Addition and Subtraction 2. Multiplication and Division 3. Exponents and Logarithms Also recall that every operation has another one that can ‘undo’ it or ‘cancel’ it out. Addition and Subtraction are inverse operations Multiplication and Division are inverse operations Exponents and Logarithms are inverse operations This brings us to the first rule of algebra: Algebra Rule #1 ‐ If you want to get rid of something in an equation, you just use its inverse operation. For example, if you add 3 to something, you can ‘undo’ this by subtracting 3. If you multiply something by 12 you can ‘undo’ this by dividing by 12. *************************************** Side Note ******************************************* When you start getting into more complicated topics, you will start doing things to numbers that are outside of the 3 groups listed above. Even so, you will find that each operation you come across will have its own inverse operation. Most of the time anything you do to a number can be undone by doing the inverse operation (there are cases with zeroes and other things that get a little sticky but for the most part this is always true). So anytime you learn a new operation, always ask yourself, “what operation will undo this?” *******************************************************************************************
Now that you get the basic idea of operations, we need to talk about this symbol: = This is a HUGE deal! The equal sign is one of the most powerful symbols in all of math and yet it is often the sign that is most commonly misused and misunderstood. The first and most important thing to remember about this symbol is that
YOU CAN ONLY USE = IF TWO OR MORE THINGS ARE ACTUALLY EQUAL!!! Here is an example. This is one of the most common mistakes that math students at all levels make, so let’s clear it up now.
Question: Calculate 2 + 5 − 3 + 7 − 1
Answer:
2 + 5 = 7 − 3 = 4 + 7 = 11 − 1 = 10
The answer given ends up at the right number but this is totally wrong!!! Read very carefully what this answer actually says. Let’s break it apart:
2 + 5 = 7 − 3 = 4 + 7 = 11 − 1 = 10
7 = 4 = 11 = 10 = 10 WHAT?!?! That is not true at all!! Be careful how you use your equal signs. Never use them to connect things unless they are actually equal. This is similar to writing a really long run‐on sentence in an essay. If you have separate ideas then write them as separate statements. The equal sign can be thought of like a scale that is perfectly balanced. Whatever quantities you have on the left must be the same weight as whatever is on the right. If I put 8 on one side of the equal sign and x+3 on the other side they have to weigh the same amount. So you can ask the question, “What plus 3 will be the same as 8?” The answer of course is 5. This brings us to our second rule of algebra: Algebra Rule #2 – If you are working with an equal sign then you can add, subtract, multiply, divide, or do any operation you want to one side as long as you do the identical operation to the other side. Any time you change something the scale always has to stay balanced.
8
x+3
Lets revisit that example x + 3 = 8 . We can solve this without really trying since we know that 5 + 3 = 8 . Even so, let’s look at the formal steps that are being used to solve this. The Goal: Get x by itself. In other words, write an equation that starts with x = So how can I manipulate the equation x + 3 = 8 so that it instead starts with x = on one side? We need to get rid of that 3. If you look back at the first rule it said that we can ‘undo’ or ‘cancel’ something by doing its inverse operation. In this case we are adding that 3 so we will do the opposite….subtract 3.
x + 3 = 8
−3
so we have a new equation:
x = 8
Wait a minute, so you are saying that x = 8? That can’t possibly be true since 8 + 3 does not equal 8! You are absolutely right! THIS EQUATION IS WRONG!!! We forgot about rule #2: we have to keep everything balanced. So if we are going to subtract 3 from the left side, we have to subtract 3 from the right side too!
The correct process will look like the following: Problem: solve x + 3 = 8
(in other words, re‐write this so it looks like x = ) Rule #1: Get rid of things by doing the inverse operation Rule #2: If you do something to one side you have to do it to the other
x +3 = 8
so we get a new equation
x = 5
−3 −3
The end! Now we have an answer that actually works. 5 + 3 is the same as 8! Ok, so what…I could have done that in my sleep. I don’t need to write out all those steps.
You are right, but what if you have to solve
(5 x − 3) ⋅ 4 = 472 ? Can’t really do that in your head so you need to −13
know how the process works so you can apply it to tougher situations. What we just solved above is an example of a one‐step equation. It only took us one step to solve. That step was subtracting 3 from both sides.
Now let’s apply this to two‐step equations. For our first example, let’s try 4 x + 7 = 19 . Remember our two rules: Rule #1: To get rid of something, just do its inverse operation Rule #2: Anything you do to one side of the equation has to be done to the other as well to keep things balanced.
Problem: solve 4 x + 7 = 19
(in other words, re‐write this so it looks like x = ) To get x by itself we need to do 2 things: get rid of the 4 and get rid of the 7 (hence, the two‐steps). Does it matter which one you do first? No, not really as long as you follow the rules. Let’s start by getting rid of the 4 first and see what happens: The 4 is being multiplied so to ‘undo’ it we will divide by 4. Remember to do it to everything to keep it equal!
4 x + 7 = 19 Æ
4 x 7 19 + = Æ 4 4 4
x +
7 19 = 4 4
Eeeww. Now I have to deal with fractions? They way we chose to do it, yes…we ended up with fractions. That’s not all bad though. At least they have the same denominator which makes them easy to add or subtract. Let’s keep pushing through it though and see what happens. So we have removed the 4 by dividing, now we must remove that
7 so we end up with x = . 4
That number is being added, so the opposite will be to subtract it. Don’t forget to do it to both sides!
x+
7 19 = 4 4
Æ
19 7 12 7 19 − = = 3 = Æ x = 4 4 4 4 4 7 7 − − 4 4 x +
We did it! So x = 3. But there is an easier way so you don’t have to use fractions all the time.
Think back to what gave us that fraction in the first place. It happened when we chose to divide by 4 first. We had to divide everything by 4 to keep it balanced so we ended up with
7 19 and . 4 4
What would happen if we changed the order that we did things? Let’s do the same problem, only this time we will remove the 7 first and then the 4. So if we look at the 7, what operation is involved? If you guessed addition you are right. So what’s the thing that cancels out addition?......Subtraction! So let’s subtract 7 from both sides of the equation.
4 x + 7 = 19
+ 7 = 19 −7 −7
Æ 4 x
Æ 4 x
= 19 − 7 = 12
That takes care of our first step. The second step is to remove the 4. So what operation is involved with the 4?.....Multiplication! And the thing that cancels out multiplication? Division!! So let’s divide everything by 4:
4 x = 12
Æ
4x 12 = 4 4
Æ
x
=
12 = 3 4
Same answer! Our x still equals 3. That’s good news! Plus, notice how this time you didn’t have to add and subtract fractions. This leads us to our third and final rule for algebra. Algebra Rule #3 – It makes no difference which values you decide to remove first as long as you follow rule #1 and rule #2. BUT…If you want to make your life easier, start by removing the ‘weaker’ operations first. (addition/subtraction then multiplication/division then exponents/logarithms then groups of terms) Wait a minute, this order sounds very familiar….where have I seen this before? It’s the order of operations!! Well…kind of. Same order, only you are doing it backwards. Of course this is just a general rule and as you start getting into bigger and tougher equations there might be a different order that you find easier. All that matters is that you follow the first 2 rules and if you can save yourself a little time by using rule 3 then that’s great. I’m still stuck on the order of operations thing. What does that have to do with solving equations? Think about what you are doing when you solve equations. You are not trying to do the operations that are in front of you. Instead you are DOING THEIR OPPOSITES!! You are trying to ‘undo’ the operations so doesn’t it make sense that you should go through the order of operations in reverse?
Let’s try another example using all three of our rules. How about 9 x − 13 = 50 ?
Problem: solve 9 x − 13 = 50 (in other words, re‐write this so it looks like x = ) This is a two‐step equation because we have two things to get rid of: the 9 and the 13. If we remember rule 3, we can make our work easier if we start with the weakest operations first. There are two operations: Multiplying by 9 and Subtracting 13. The weaker of the two is the subtraction so let’s take care of that first by doing it’s opposite…adding 13.
− 13 = 50
9 x − 13 = 50
Æ
9 x
(Do it to both sides)
+13 +13
Æ 9 x
= 50 + 13 = 63
Now the 9. The x is being multiplied by 9 so we cancel it out by dividing by 9.
9 x = 63
Æ
9x 63 = 9 9
Æ x =
63 = 7 9
And we’re done! We have x = 7. These examples are pretty easy…what if the answer doesn’t come out with such a nice whole number? This is a very good possibility, so let’s take a second and remember how we would check our answer for something like this. Take the previous example where x = 7. Our original equation was 9 x − 13 = 50 . Remember this means that 9 x − 13 must be the same as 50. So if we put in 7 for x it should come out to be 50. 50 50 9(7) − 13 Æ 50 Æ 63 − 13 50
It is always a good idea to check your answers since it’s really easy to do and always guarantees you have the right answer!!
We’ll write two more full problems to look through without any explaining. See if you can follow the steps and notice where we are using the three rules.
Problem: solve 43 − 6 x
= 25
43 − 6 x = 25
Æ
− 6 x = 25 − 43 − 43
− 6 x = −18
Æ
Solution: x
43
− 6 x −18 = −6 −6
Æ −6 x
= 25 − 43 = −18
Æ x
=
−18 = 3 −6
= 3
Problem: solve −19 +
x = −14 7
−19 +
x = −14 7
Æ
x = −14 7 + 19 + 19 −19 +
Æ
( 7 ) ⋅
x = − 14 + 19 = 5 Æ 7
Æ x
x = 5 7
x = 5 ⋅ (7) 7
Solution: x
= 35
= 5 ⋅ 7 = 35
Did you see the steps? You can see how these types of equations can start to get confusing very fast. You can do this with negatives, fractions, decimals….anything. Just be careful if the answer is not obvious right away. Check your steps carefully and don’t forget to follow the rules: Rule #1: To get rid of something, just do its inverse operation Rule #2: Anything you do to one side of the equation has to be done to the other as well to keep things balanced Rule #3 – It makes no difference which values you decide to remove first as long as you follow rule #1 and rule #2, but if you remove the ‘weaker’ operations first, it will often be easier.
AND…..DON’T FORGET TO CHECK YOUR ANSWERS! Plug your answer back into the original equation and see if the two sides actually come out the same. Good luck and happy solving! www.mathmadesimple.org