Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
With practice, linear equations can be straightforward to solve. In this lesson, we'll define linear equations and learn how to solve them. We'll look at multiple practice problems and walk through solving each one.
X Marks the Spot
X. X is everywhere. It's the checkbox on the user agreement we never read. It marks railroad crossings and superhero uniforms. And, it's at the end of every pirate map. In all these examples, x stands for something. For example, on that pirate map, x is where you'll find the buried treasure. In algebra, x is sort of like a marker for treasure, if by treasure we mean a number. That's because x is a common variable. And, a variable is a symbol used to represent a number.
Here we're going to solve for x in linear equations. It's just like following a pirate map, where we follow the clues until we know where x is. Fortunately, with linear equation solving, you're much less likely to lose a leg or an eye. And, scurvy is very rare. First, let's do a quick review of what these equations are.
A linear equation is simply an algebraic expression that represents a line. These equations commonly contain one or two variables, usually x or y. These are called first-degree equations because the variable's exponent is always one.
We won't see anything like x^2 or x^3. Those may get you lines like what you'd actually see on a pirate map. But, our pirate map has straight lines. It's much easier that way. You also won't see things like x times y, x over y or the square root of x. That's for pirates who travel through time and space.
Oh, and the linear equations we'll be solving here only have one variable, not two or more. I mean, if we had both x and y, how do we know which one has the treasure and which one is a trap? Pirates are big into traps.
Solving Linear Equations
As I mentioned before, to solve one of these equations, we're trying to solve for x. If you have x - 4 = 10, then 10 isn't where the treasure is. It's where x - 4 is. And, that's just not what we need.
To solve this equation, we need to get x alone on one side of the equation. Here, we do that by adding 4 to each side. That gets us x = 14. So, that's our treasure. I know that's a modest haul, but it was a basic equation.
For all of these equations, we'll always just do whatever we can to isolate x. If we have 2x = 6, we divide by 2 to get x = 3. Just keep your focus on x.
Now that we know what to do, let's go after some serious treasure. Let's start with x + 2 = 9. This looks like the first one we saw. This is like Pirate Treasure Hunting 101. Let's subtract 2 from both sides to get x = 7. It's another small treasure, but it is ours.
What about 3x - 4 = 11? Let's add 4 to each side. Then divide by 3. We get x = 5. That's a little more gold. We could buy ourselves a parrot.
Here's another: 4x - 9 = 2x - 3. We have xs on both sides, so we need to move them around. Let's subtract 2x from both sides. Now add 9 to both sides. Then divide by 2. That's x = 3. I think we can get a pretty sweet sword with that.
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Okay, now let's try x/4 + 2 = 5. First, subtract 2 from both sides. And, how do we get rid of that 4? We multiply both sides by 4. So, x = 12. Maybe we should put some of this gold in our Pirate College Fund. You know, pirates don't do that well in school. They're always in the high Cs.
Okay, that was bad. Let's go after more treasure. What about 2(x + 4) = 22? We first need to distribute that 2 across the x + 4. We get 2x + 8. Next, subtract 8 from both sides. Then divide by 2. So, x = 7. Maybe we can buy a better pirate joke book with that.
Here's one: 6x - 1 - x = 26 - 4x. Let's combine all those xs. 6x - x is 5x. Then we add 4x to both sides to get 9x over here:
Adding 4x to both sides gives you 9x on the left side of the equation.
Add 1 to both sides to get 9x = 27. Divide by 9, and x = 3. Now we're getting some gold.
Let's try a tougher one: 3x/4 + 2/3 = 5/6 + x. Oh, man. What do we do here? We need the lowest common multiple of 4, 3 and 6. That's 12. So, we multiply both sides by 12. That gets us 12(3x/4) + 12(2/3) = 12(5/6) + 12(x). That last one is just 12x. What about the others? 12(3x/4) is 36x/4. That reduces to 9x. 12(2/3) is 24/3, which is 8. And 12(5/6) is 60/6, or 10. Okay, 9x + 8 = 10 + 12x. Let's subtract 12x from both sides to get -3x over here:
Subtracting 12x from both sides gives you -3x on the left side of the equation.
Then subtract 8 from both sides to get 2 over here:
Subtracting 8 from both sides gives you 2 on the right side of the equation.
Finally, divide by -3. So, x = -2/3. Now that's a treasure. We could buy our very own pirate ship with that.
In summary, being a pirate is all about finding the right parrot. Wait, no. That's not what we learned here. We learned about solving linear equations. A linear equation is an algebraic expression that represents a line. We focused on single variable equations, where we try to get x alone on one side of the equation. Once we find x, we have our treasure!
You should be able to solve linear equations with a single variable after watching this video lesson.
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