3(x-5) = -7x + 12
If you are like me, you panic when this equation is thrown in front of you; however, HAVE NO FEAR! There is a simple way to solve this crazy equation. Let's do it step-by step. Each picture below illustrates a step in the process of solving this equation with a variable on each side.
Here is our original equation. I learned to set it up like a table, to keep my steps organized.
To begin, the 3 must be distributed to everything in the parentheses. 3 multiplied by X is 3x, and 3 multiplied by -5 is -15. Copy down the rest of the equation.
This cannot be solved until all X's are on the same side of the equals sign. I could either subtract 3x from the left, or add 7x to the right. I like working with whole numbers, so I chose to add seven. Copy down your new equation.
Now that all X's are on the same side of the equal sign, we need to isolate it. This happens by performing the opposite of subtraction, which is multiplication. Add 15 to both sides of the equation remembering that anything that happens to one side of an equation, must happen to the other.
Good work. We are close to finished, but we haven't isolated our variable. 10 is being multiplied by X here, so we will perform the opposite and divide by 10 on both sides of the equation.
CONGRATULATIONS MASTER OF CRAZY EQUATIONS!!!
Thanks Alison, very nicely done. I really like how you engage your reader by showing some empathy, I think that helps them feel more comfortable continuing to read and work through this problem. And, like last time, your visuals are great and your explanation is well thought-out and clearly done.
ReplyDeleteA few suggestions (again, in the spirit of getting even better):
1. You say you can either "subtract 3x from the left, or add 7x to the right." A couple of things here - you are actually either subtracting 3x from both sides, or adding 7x to both sides, so be careful with that wording. Second, I think it might be nice to tell your reader why you are doing that step. You tell them you need to get all the x's on the same side, but I think being a little more specific about how subtracting 3x or adding 7x does this would be helpful (remember, you have to assume they know nothing about how to do this).
2. You said "I like working with whole numbers, so I chose to add seven." I think you meant "positive numbers," correct?
3. You said "by performing the opposite of subtraction, which is multiplication." I know what you meant :-), but proofreading is always a great idea, especially when publishing online.
4. Did you consider talking about how they could check to see if their answer was correct?
All in all, another excellent job. Keep up the good work!
I like the pictures Alison, it really helped me understand what you were doing. I think I'll recommend my students try the same process.
ReplyDeleteAs a mentor of new teachers, I am going to share your site with my math teachers. So many students struggle with these concepts. I really like your use of color to isolate each step of the process. It looks like your teacher gave you some excellent feedback. Thank you for sharing your math knowledge and learning with the world. :-)
ReplyDeleteI've done thousands of problems like this and I still enjoy hearing and sharing our strategies for solving them. If you really want to become a ninja equation solver, make sure you can answer these questions:
ReplyDelete1. You said, "anything that happens to one side of the equation, must happen to the other." Yet you started the problem by performing the distributive property on only one side. Why didn't you have to do the distributive property on both sides? (If you think it's just because there's no parentheses/grouping on the right on which to do the distributive property, then keep thinking! The answer is much more important than that.)
2. When you added 7x to both sides of the equation, you correctly showed 10x on the left. But what did you create with the x's on the right? What's special about that number? Similarly, what did you get on the left when you added 15 to both sides? What mathematical property makes you want that number, even though you're probably not focused on it?
3. In Step #4 you divided both sides by 10. Again, you correctly got 27/10 on the right. But what do you get when you make 10/10 on the left? What's special about that number? How is it similar to the special number in my question #2? How is it different?
4. How does your first equation, 3(x-5)=-7x+12, compare to your last equation, x=27/10? Are they different? Are they the same? What about the equations you wrote in between?
I can already tell you're good at solving equations, but don't be surprised if you have a little trouble making sense of the answers to my questions. That's okay -- developing ninja-level equation solving skills takes time!
I really like the idea of setting up the table -- does help keep it organized. I will share the the idea with with my students.
ReplyDeleteThanks, Mrs.D - the Rocket City