# Solving Equations by Adding and Subtracting

This lesson will introduce what variables and equations are and how we can solve for an unknown quantity using the addition property of equality or subtraction property of equality. We discuss what equations are and how to use them to solve problems as well as how to isolate a variable and thus solve an equation using addition or subtraction.

So without further ado, read through the slides below to get a feel for how to solve equations using addition and subtraction!

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Solving Equations by Multiplying and Dividing

This lesson is a video generated from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. You may also head over to my YouTube channel to see the slideshows in video form there!

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

• Solving equations
• By adding or subtracting
• Objectives
• By the end of this lesson you should feel comfortable:
• Identifying and using variables
• Utilizing the addition and subtraction properties of equality to isolate variables
• unknowns
• In the real world of mathematics, often times we are asked to solve for some unknown quantity rather than just given a problem in a straightforward manner.
• For example:
• You have a budget of \$50 on a shopping trip and have already spent \$35. How much do you have left to spend?
• You see an item that weighs 30 lbs more than you can lift. If you can lift 120 lbs, how much does the object weigh?
• Both of these may seem similar to word problems you have solved before:
• unknowns
• There is another way to write these problems, however:
• Both of these match our situations better if we think critically.
• In our first problem, we have spent \$35 and want to know how much more we have until we reach our limit of \$50.
• In the second, we know that however much the object weighs, we can lift 30 lbs less than that to reach our 120 lbs max.
• Written like this, our word problem can translate more literally into a solvable number sentence.
• The good news is that we will solve these exactly how we set them up in the first place! But bear with me until then to learn something new!
• Variables
• In math, we don’t just leave equations looking like this:
• Instead, we have something called a variable.
• A variable is a letter that represents an unknown, or able to vary, quantity.
• See “able to vary” and “variable.” Mathematicians are very lazy and so name things in a rather obvious way most of the time!
• And our most common variable is “x.”
• Notice how it looks a lot like a multiplication symbol?
• That is why in math a number next to parentheses is another way to write out multiplication in a problem.
• Equations
• Now we finally have two equations.
• An equation is a number sentence containing a variable and an equal sign, thus equating both sides to one another.
• In our first example, 35 + x is equal to 50. They are the same value.
• Similarly, x – 30 is equal to 120, again saying the values are the same.
• We are not yet sure what value x represents in either case to ensure that the statements are true, but we can find out!
• Additon Property of Equality
• Let’s actually begin with our second example. How would we go about solving this equation for x?
• We need to isolate x first, or get it by itself. This means getting rid of the 30 from the left side.
• This is where our first property comes in.
• The addition property of equality says that we can add whatever we want to one side if we also do so to the other side!
• Check it out:
• Addition property of equality
• Imagine our equation is like a scale with a balance centered under the equal sign.
• If we were to add to only one side, the scale would become unbalanced…
• As long as both sides remain equal, then our equation is balanced.
• And this is exactly what the addition property of equality tells us!
• So we should add 30 to the other side, the left, as well.
• Hmmm something is happening here, though…
• Addition property of equality
• We are left with this:
• Aha, but adding 0 to anything just leaves itself, right?
• We know this is true because that is exactly what the identity property of addition tells us!
• So………
• x has been isolated. It is by itself.
• In fact, now our equation very simply tells us that x = 150.
• Addition property of equality
• Ok, a lot just happened, so let’s calm down with another example using the addition property of equality:
• Perfect. So let’s think about all we really did in the previous problem.
• Why did we add 30 to both sides? Because we saw -30.
• -30 and 30 are inverses, so we can really just imagine them canceling out.
• What is the opposite of subtracting 10?……Adding 10!
• Make sure to do the same to both sides so they stay equal.
• Addition property of equality
• Now, instead of thinking that -10 + 10 = 0 and that x + 0 = x, just imagine the 10’s canceling each other out like so:
• Then we are just left with our solution after adding on the right!
• Three steps:
• Add the opposite of what you are subtracting to both sides.
• Cancel the opposites to get x by itself.
• Add the other side to get your answer!
• Subtraction Property of equality
• With all the building blocks we have, figuring out how to solve our first example equation should be much quicker.
• As you can see, this time we are adding. And judging by the title of this slide, it might be wise to consider applying the subtraction property of equality.
• Much like its counterpart, the subtraction property of equality states that we may subtract any number from one side as long as we subtract the same number from the other side.
• Subtraction property of equality
• Let’s try following our three steps again, modifying very slightly.
• First, we still need to add the opposite of whatever is next to x, but the opposite of 35 is -35.
• Adding a negative is like subtracting, remember? (They go the same way on the number line, to the left.)
• And we do the same thing to both sides to remain balanced.
• Second, we cancel our opposites.
• Then finally we add down to have x isolated!
• conclusion
• A variable is a letter that represents an unknown value, usually the x.
• An equation is a number sentence containing at least one variable and an equal sign.
• The addition property of equality states that you may add any number to one side of an equation as long as you add the same number to the other side.
• The subtraction property of equality states that you may subtract any number from one side of an equation as long as you subtract the same number from the other side.
• Utilizing inverses, we isolate a variable by removing everything else from that side of an equation so that we can read a solution “variable = solution”