Lesson 3 Skills Practice Write Two Step Equations
Hey there, math adventurers! Ready to tackle Lesson 3 Skills Practice? We're diving into the super-duper fun world of writing two-step equations. Don't worry, it's not as scary as it sounds. Think of it like solving a little mystery, where the variable is your sneaky suspect and the equation is the clue you need to crack the case!
So, what exactly is a two-step equation? Well, it's an equation that, surprise, surprise, requires two steps to solve. Mind. Blown. Right? These equations usually involve a variable (that's our X or Y, or whatever letter we decide to be a detective) and two operations. We're talking addition or subtraction, and multiplication or division. It’s like a mini-puzzle that tests your logic and your ability to follow instructions.
Before we jump into writing them, let's do a super quick recap of what we’ve already learned. Remember one-step equations? Those were like the warm-up laps. If you had something like x + 5 = 10, you just needed to subtract 5 from both sides to get x all by its lonesome. Easy peasy, lemon squeezy. We’re building on that foundation, so if you’re feeling a little rusty on those, a quick peek at your notes might be a good idea. No shame in a quick refresh – even Sherlock Holmes consulted his notes sometimes!
Now, let’s get to the juicy part: writing these two-step equations. This is where we translate word problems into the language of math. Imagine you're a spy, and the word problem is your coded message. Your mission, should you choose to accept it, is to decode it into a mathematical equation. Exciting, right?
Let’s start with a classic. Imagine this: "Sarah bought 3 notebooks and a pen for $2. If the total cost was $8, how much did the pen cost?"
Okay, let’s break this down. First, we need to identify what we don’t know. What are we trying to find? In this case, it's the cost of the pen. So, let's assign a variable to represent the cost of the pen. We can use 'p' for pen, or if you're feeling fancy, 'x' is always a safe bet. Let's go with 'p' for a little thematic fun. So, p = cost of the pen.
Now, what information are we given? Sarah bought 3 notebooks. Hmm, the problem doesn't tell us the cost of one notebook, but it does tell us the total cost of the notebooks was... wait a minute. Rereading the problem, it says "Sarah bought 3 notebooks and a pen for $2." This wording is a little tricky! It implies the pen cost $2, and then there's the notebooks. Let's adjust our word problem slightly to make it a clearer two-step scenario. Let's say: "Sarah bought 3 identical notebooks and a pen for $2. The total cost was $8. How much did each notebook cost?"
Okay, that's more like it! Now we have a clear unknown: the cost of each notebook. Let's assign our variable here. Let n = the cost of one notebook.
What did Sarah buy? She bought 3 notebooks. So, the cost of the notebooks would be 3 times the cost of one notebook, which is 3n.
She also bought a pen for $2. So, we have the cost of the notebooks (3n) plus the cost of the pen ($2). That gives us a grand total of 3n + 2.
And what was the total cost? The problem tells us it was $8. So, we can set our expression equal to the total cost:
3n + 2 = 8
Voilà! We've successfully translated that word problem into a two-step equation. See? Not so terrifying, right? It’s like piecing together a puzzle, one word at a time. You identify the unknown, you figure out how the knowns relate to it, and then you build your mathematical sentence.
Let’s try another one. Imagine this scenario: "David is saving up for a new video game that costs $60. He already has $15 saved and he earns $5 each week from his allowance. How many weeks will it take him to save enough money for the game?"
Alright, detectives, what's our mission here? We need to find out how many weeks it will take David to save up. So, let's assign our variable to represent that. Let w = the number of weeks.
What do we know? David already has $15 saved. That's a good head start! He also earns $5 each week. So, for every week that passes, his savings increase by $5. If 'w' is the number of weeks, then the total amount he earns from his allowance will be 5w.
So, his total savings will be the money he already has plus the money he earns. That’s 15 + 5w.
And what's the goal? He needs to save $60 for the video game. So, we set our expression equal to the target amount:
15 + 5w = 60
Boom! Another two-step equation, written and ready to be solved. See how we're building it up? Identify the unknown, represent it with a variable, then use the given information to construct the equation. It’s like giving the math world a recipe, and the equation is the finished dish!
Let's think about some common phrases that pop up in word problems and what they translate to in math. This is like learning the secret codebook of math.
"More than," "added to," "sum of" usually means addition (+).
"Less than," "subtracted from," "difference" usually means subtraction (-). Be careful with "less than" – it can sometimes reverse the order! For example, "5 less than x" is x - 5, not 5 - x. It’s like saying you have 5 apples fewer than someone else – you subtract 5 from their amount.
"Times," "product of," "multiplied by" means multiplication (× or ).
"Divided by," "quotient of," "ratio" means division (÷ or /).
Let's try to write an equation from this phrase: "Six more than twice a number is 20."
First, what's our unknown? It's "a number." Let's call it x.
Now, let's break down the phrase: * "Twice a number": This means 2 times our number, so 2x. * "Six more than...": This means we add 6 to whatever came before it. So, it's 2x + 6. * "...is 20": The word "is" almost always means equals (=). So, it’s = 20.
Putting it all together: 2x + 6 = 20.
See? You're basically dissecting the sentence and replacing the words with their mathematical equivalents. It’s like building with LEGOs, but with numbers and symbols!
Let’s do one more, just to be sure. How about: "One-fourth of a number, decreased by 3, equals 7."
Unknown: "a number." Let's use y this time. Why not?
Breaking it down:
"One-fourth of a number": This means (1/4) times our number, or (1/4)y. You could also write this as y/4.
* "...decreased by 3": This means we subtract 3. So, we have (1/4)y - 3.
* "...equals 7": This means = 7.
So, the equation is: (1/4)y - 3 = 7 or y/4 - 3 = 7.
Now, I know what you might be thinking: "But what about solving these things?" Well, the beauty of this lesson is that you're mastering the writing part. Once you can write the equation correctly, solving it becomes the next logical step, and you've already got the skills from dealing with one-step equations. It's all about isolating that variable!
Remember, the goal when solving is to get the variable all by itself on one side of the equals sign. We do this by using inverse operations. Addition undoes subtraction, subtraction undoes addition, multiplication undoes division, and division undoes multiplication. And the golden rule of equations? Whatever you do to one side, you must do to the other side. It’s like keeping a perfectly balanced scale. If you add weight to one side, you have to add the same weight to the other to keep it even.
For 3n + 2 = 8, we would first subtract 2 from both sides to undo the addition. That gives us 3n = 6. Then, we would divide both sides by 3 to undo the multiplication. And voilà, n = 2!
For 15 + 5w = 60, we would first subtract 15 from both sides to undo the addition. That gives us 5w = 45. Then, we would divide both sides by 5. And guess what? w = 9!
For 2x + 6 = 20, subtract 6 from both sides: 2x = 14. Then divide by 2: x = 7.
And for y/4 - 3 = 7, add 3 to both sides: y/4 = 10. Then multiply both sides by 4: y = 40.
See? The solving part often falls into place once the equation is written. This lesson is all about building that strong foundation of understanding how to translate the real world (or at least, the world of word problems!) into the precise language of mathematics. It’s a superpower, really!
So, next time you see a word problem, don't groan. Smile! See it as an opportunity to put on your detective hat, grab your mathematical magnifying glass, and uncover the hidden equation. You’ve got this! You’re building skills that will help you not just in math class, but in figuring out all sorts of things in life. Every equation you write and solve is like a little victory, a step closer to understanding the world around you a bit better. Keep practicing, keep exploring, and remember – you’re doing great!