Weekly Math Problem #61

A Hybrid Equation. Yeah...I don't know what else to call it. 🤷🏿‍♀️ I was going through an old math worksheet for the NYS Algebra 2 Regents Examination from JMAP, when I stumbled across what is now this week's problem. 🤔 I don't recall solving anything like this in the past...but I could definitely be wrong. If I have, then it was a very long time ago. This is why I chose the problem. Based on the math I know, I am attempting it this week! Why don't you join me?

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to solve equations involving absolute value expressions
        ✔️     How to solve equations involving quadratic expressions

     

WMP! #61 want us to...

Happy solving!

Check back on Saturday, October 30th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

Weekly Math Problem #59

Venn Diagram. When I think about it, I learned about a Venn diagram decades ago without really knowing anything about Venn. I used to think, "What is a Venn?", instead of thinking, "Who is Venn?" 🤷🏿 I can't recall ever looking up Venn, so it looks like I am doing this for the first time. Venn is John Venn. John Venn was an Englishman born in 1834 who became a mathematician and philosopher. (This information was pulled from Wikipedia : John Venn.) John Venn was also part of the #beardgang. 😁 Anyhow, I used to find Venn diagrams confusing, when first learning them. Now, I definitely get it. 👍🏿

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to understand a Venn diagram
        ✔️     Simple probability

     

Ready or not, here is WMP! #59:

Happy solving!

Check back on Saturday, October 16th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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I don't know how you feel about Venn diagrams, but I definitely understand why working with them can be confusing at times. I started this problem by identifying the cardinality of each set. There are three sets represented in this problem--R, J, and Zeta (the universal set). The cardinality of a set is the amount of elements in the set. In this problem, algebraic expressions are used to represent some of the cardinalities. "n(set_name)" is the notation used for cardinality.

Before I can move forward to find the probability of interest, I need to know what the exact cardinality of set R is. This requires me to solve for x. I can use both representations for the cardinality of set Zeta to do this.

Here are the exact cardinalities for the three sets:

Now, I have the information I need to find the probability of interest.

The probability that a randomly chosen car is 7/30.

▪️ Did you find the probability??

▪️ Let me know what you thought about this week's problem in the comments section. 

Thanks for solving with me this week!

Up next...WMP! #61. 👍🏿

Cheers!

The Younge Lady

Weekly Math Problem #55

System of Linear Equations. I know it has been a while since the last problem (of which I still owe you my solution 🥴...it's coming, it's coming). Please don't disown me. When life throws you curveballs, you learn how to catch 'em. ⚾️🤷🏿‍♀️ ...And, I'm still learning how to catch. 

So, to get my feet wet, I'm starting with the problem below. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     Methods for solving a system of linear equations
        

WMP! #55 says to...

Happy solving!

Check back on Friday, September 17th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady ( I'm gonna need it this year. 😆 )

✏️📓 Solution Time! 📓✏️

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I went through my blog to see which WMPs already involved this week's topic. So,  check out WMP! #4 and WMP! #28 to how I solved those systems. WMP! #4 is more similar to this week's problem than WMP! #28 is. 

Just as a reminder, you have multiple options to use when deciding how to solve the system--addition/elimination method, substitution method, and Guassian elimination (if you're feeling fancy 😉). I am going to go ahead and do the substitution method. The first thing I do is label my equation. This helps me to keep track of things.

Next, I need to examine the equations to see which methods I want to move forward with. For the fun of it, I've decided to do the substitution method. Equations (A) and (B) both possess variables with "1" coefficients. Since equation (A) has a variable with a positive 1 coefficient, I'll just work with that. 

I isolate the variable and label the rearranged version of the equation:

Now, I will substitute this equation into the other two original equations (B) and (C). This allows me to reduce the variables--going from three variables down to two. 

After substituting and simplifying, I have a system of two linear equations in two variables. I examine this system and proceed with the substitution method again. Repeating a similar process from above.

Once this round of substitution is complete, I have come up with a value for one of the variables. Moving forward with with backward substitution...lol...I am able to find the values for the remaining variables and present my solution.

As you may know with mathematics, getting an answer doesn't mean that it's correct. So, I checked my solution in the original system to make sure that it works completely.

Thanks for hanging with me this week. It feels good to get back into the swing of things. 

▪️ What is your preferred method for solving a system like the one from this week's problem??

▪️ Let me know what you thought about this week's problem in the comments section. 

Thanks for sticking around and solving with me this week!

Up next, WMP! #56. 👍🏿

Cheers!

The Younge Lady

Weekly Math Problem! #24

Trigonometric Equation. The last time I had a trigonometric equation on here was back in June, WMP! #8. (Be sure to check it out. 😉)  That trig equation didn't have any of the functions squared. This week's problems involves a squared function and a function with a double angle. I know, I know... Why, Lori?! Because it looked fun to solve. 🤓

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️ Double angle formulas
        ✔️ How to solve quadratic equations
        ✔️ How to isolate the angle in a trigonometric function

By the way, you'll definitely need at least a scientific calculator for this one. Here is WMP! #24:

Happy solving!

Check back on Friday, October 23rd for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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This problem is a quadratic trigonometric equation. What makes it "difficult" to solve is the trigonometric function with the double angle. How do we deal with that? Use a double angle formula to replace it in the equation. Let me show you...

                                                                       

As you can see, switching out the double angle trigonometric function for an equivalent one helped a great deal. It allowed for the original equation to be reduced to a simpler quadratic equation that could be solved with ease...and a calculator...lol 😅.

 

▪️ Did you use the same double angle formula I used to solve?? 

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

See you next week for WMP! #25

👍🏿

Cheers!

The Younge Lady

Weekly Math Problem! #23

Radical Equation. The last time I had a radical equation on here was back in February, when I showed a sample WMP! It's time for another one, so here we go.

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️ How to deal with radical equations
        ✔️ How to solve quadratic equations
        ✔️ How to solve linear equations
        ✔️ How to substitute values for checking potential solutions

WMP! #23 says:

Happy solving!

Check back on Friday, October 9th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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I did a similar problem type when I posted my WMP! Sample Edition problem back in February. This time, the radical equation involved a square root within a square root. Here's my solution:

As you can see, this problem yielded one extraneous solution and one valid solution.  

 

▪️ How do you feel when you saw this problem?? 

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

See you next week for WMP! #24

👍🏿

Cheers!

The Younge Lady

Weekly Math Problem! #21

Integration. So, here we are again with another integration problem. If you've done last week's problem, then you've already done some of the work needed to do this week's problem. 🤓 Wait, you haven't seen last week's problem?! Pause....check out WMP! #20. 

Last week I mentioned that partial fraction decomposition is an important technique that allows you to "simplify" some complicated rational expressions by breaking them down into smaller parts. WMP! #21 is such a problem that requires the technique of partial fraction decomposition.

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️ How to integrate rational expressions

        ✔️ Partial fraction decomposition

WMP! #21 wants us to...

Happy solving!

Check back on Friday, September 25th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Again, if you take a look at WMP! #20, you will see that it has the same rational expression as this week's problem. So, when you're given a rational expression, similar to that one, to integrate, you can see what needs to be done to carry out the integration. Here's how I carried it out:

That's all I've got! (I actually really like this problem.)  

 

▪️ How did you feel about this week's problem?? 

▪️ Did you carry out the integration differently than I did??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

See you right back here next week for WMP! #22, the last problem of September.

🤓

Cheers!

The Younge Lady

Weekly Math Problem! #20

Partial Fractions Decomposition. This topic is one I didn't hear much about until I was required to integrate certain types of rational expressions. It's an important technique that allows you to "simplify" some complicated rational expressions by breaking them down into smaller parts, so that you may more easily compute problems that involve them. This week's problem comes from the Pure Mathematics, CAPE (Caribbean Advanced Proficiency Examination) materials.

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️  How to use partial fraction decomposition

        ✔️  Simplifying rational equations

        ✔️  Working with undetermined coefficients

    

WMP! #20 says...

Happy solving! 

Check back on Friday, September 18th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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...And we're back to decompose this fraction into the components that were combined to get it in the first place. Here are the steps I took to do so:

There you have it!  

 

▪️ Were you able to decompose the fraction?? 

▪️ Comment below with your responses and let me know what you thought about this week's problem.

See you right back here next week for WMP! #21

(Now let's see if you can guess what next week's problem is going to be.)

🤓

Cheers!

The Younge Lady

Weekly Math Problem! #19

Oblique Triangles. Back to some good ol' trig. 👍🏿 Different than special triangles, oblique triangles don't really have ways to come up with any of the values for the angles or legs of the triangles by following "shortcuts". You pretty much have to rely on the Law of Sines or Law of Cosines to get the answers to the problems. 

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  When/How to use the Law of Sines

    ✔️  When/How to use the Law of Cosines

    ✔️  How to find the area of an oblique triangle

    

By the way, you'll need a scientific calculator to help you in computing.

WMP! #19 says...

Happy solving! 

Check back on Friday, September 11th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Where to start? Well, there is not enough information to solve for any of the unknown angles, B or C. So, we'll solve for the missing side, a. One thing to notice is that the given triangle is not a right triangle, therefore, we cannot use the Pythagorean theorem. That's okay. We'll use something that looks a lot like it--the Law of Cosines formula. We'll use the one that fits our given information.

Now that all the sides are known, we have enough information to solve for the remaining angles, B and C. You have the option to use the Law of Cosines formula again, or you can use the Law of Sines. There's a catch...🤨. Sorry. 🤷🏿 . If you choose to use the Law of Sines, then you will run into a problem solving for angle B. Why?? "A triangle with two sides and a consecutive angle is the ambiguous case of the Law of Sines. In this case, you may have one, two, or no solutions, (Pre-Calculus Workbook for Dummies, Gilman, et al., 2009, p. 179)." Using the Law of Sines to solve for angle C is just fine. If you continue to solve for the remaining angles using the Law of Cosines, you also be fine. What did I do? I used the Law of Sines to solve for angle C.

Now that a second angle has been found, we can easily use subtraction to solve for the last angle:

It's done...the triangle has been solved for! Here's the triangle summary:

The last thing the needs to be done is to find the area of the triangle. Since the height of the triangle cannot be easily seen or found, we will not be using the traditional area formula of a triangle. Here's another formula to find the area of an oblique triangle:

The area can also be found by replacing the sides and angle, accordingly, in the formula. Similar results will be found.

 

◾️ How did you do in solving this week's problem?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Up next... WMP! #20❗️

Cheers!

The Younge Lady