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! #38

Linear Regression. I scanned my current problems list and said, "Hmm...🤔... I don't see STATISTICS on here." I thought it would be fun to do a little linear regression so, here we are. I think STATISTICS is cool, however, there are many instances when performing the calculations can make you feel a little dizzy. 😵 This is why we're grateful for software. 🙃 

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

        ✔️     How to find the regression line using the Least Squares method
        ✔️     How to use the regression line
        ✔️     How to interpret the slope regression line

You'll have to lookup the formulas and so will I. Perform the calculations by hand and use software to check. I know the numbers a little big but, we'll be okay. (You may want to do the calculations in stages, if doing it in one sitting is too much.) Alright, here goes WMP! #38:

Happy solving!

Check back on Friday, February 19th 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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Hopefully, you're not mad at me for these calculations. 🥴 Somehow, I found this to be a little, key phrase "a little", therapeutic. Yeah, I know, I know, I'm weird...but who isn't? 🤷🏿‍♀️ I will acknowledge that mistakes can easily be made, especially if you're not paying attention. I had to check my work...a must. I entered the data in Google Sheets to do so. Let's get started shall we...

a. To find the equation of the regression line, certain calculations are needed. We'll start with squaring and multiplying the data points, as needed, organizing them in chart. That information will, then, be used to calculate the intercept and the slope of the regression line. Finally, the equation of the regression line can be written.

   

b. To estimate the blood pressure, we'll use the equation of the regression line from part a. How? Substitute the age 50 in place of x, then perform the indicated operations. The answer you get will be the estimated blood pressure, assuming no computation errors were made.

c. In our situation, the residual is the difference between the blood pressure of a 42-year old as observed in the data and the estimated blood pressure of a 42-year old as computed from the regression line equation.

d. To interpret the slope, keep in mind the context of the problem and think about how the blood pressure number changes, when the age of a person increases by one year.

I hope you found this helpful. I sure did! It was super helpful for me, actually. Part d. was my favorite part of the problem because, it required to me explain the meaning of the slope. Interpreting/explaining numbers reminds me that it isn't always about what the number is. Just as important is what the number means. This is how reports get written. This helps with decision making in certain sectors. 

But before we go, I just have one more thing to share...a scatter plot of the data from the problem with the fitted regression line.

*This plot was created using Google Sheets.*

▪️ Did you find this problem hard, annoying, or something else?

▪️ How did you feel about STATISTICS, in general?

▪️ Leave your response down below and let me know what you thought about this week's problem.

Thanks for solving with me this week!

L👀ks like WMP! #39 is quickly approaching.

Cheers!

The Younge Lady

Weekly Math Problem! #14

Matrix Operations. Ahh...matrices. Just when you thought basic math operations couldn't get more complicated...enter numbers in block form. 😅🤦🏿‍♀️ In my years of doing mathematics, I see that every area of mathematics has practical applications. The truth is, most people won't learn math at the levels where such applications exist. Also for the people who do go on learn math at those levels where these applications exist, accompanying software is also learned. Doing that type of math by hand just isn't efficient. Linear Algebra has some cool applications but, before you can learn the math needed for such applications, you have to start with the basics.  

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

    ✔️  How to perform basic matrix operations

    

Here is WMP! #14...

Happy solving! 

Check back on Friday, August 7th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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In colored fashion, here's how the matrix calculations go:

Matrix multiplication is definitely tricky but, with a keen eye, a pattern can be seen. Yeah...don't have much to say this week; it's been a hectic one. In an upcoming WMP, I'll do a problem from the math journey I took this summer. 👀 Wondering what it is? You should stick around to find out. 😃

◾️ Have you ever done linear algebra or learned about matrix theory before?? 

◾️ If so, how long ago and what was your experience like?? 

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

Next up...WMP! #15 😏

Cheers!

The Younge Lady

Weekly Math Problem! #10

Solving An Exponential Equation. Exponential equations are very different from polynomial equations (linear, quadratic, etc.) not just in the way they look, but also in behavior and methods needed to solve them. Exponential equations sometimes require a different type of "manipulation" via logarithmic properties, as opposed to what is needed to solve polynomial equation types, which do not require that. 

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

    ✔️  Logarithmic properties

    ✔️  How to solve polynomial equations

    

I present to you WMP! #10...

Happy solving! 

Check back on Friday, July 10th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Listen here...I looked at this problem an automatically assumed that all I needed to do was, somehow, get the exponential term in one spot--factoring it out--then I'll be good. Nah fam, not at all. 😕 When I tell you that wasn't it....that was not it. I needed another technique and wasn't quite seeing it. I was stumped for a bit. What I needed wasn't apparent to me because I was not used to using this technique to solve a logarithmic equation of this form. Factoring is what I needed to do...just not in the way I originally thought. Here's what I did:

I had to solve this thing by factoring a trinomial. It took me too long to see it. Once I saw it, it was solved in minutes. Even though, I didn't post it this week, I did check both solutions. None of them are to be rejected. 

◾️ Have you ever solved a logarithmic equation like the one above??

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

On to WMP! #11 ▶️

Cheers!

The Younge Lady

Weekly Math Problem! #4

System of Linear Equations. Typically, when you've encountered this topic in learning math, you've either encountered a system of two linear equations with two unknowns (from an algebra course), or a system of three linear equations with three unknowns (from a college algebra course). I don't recall ever having to solve a system of four linear equations with four unknowns, however, when I saw it I knew that I would be able to solve it. I knew what I needed to do.

Depending on what level of math you've completed, you're aware that you have a few options when it comes to solving a system of four linear equations with four unknowns. To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Methods for solving systems of linear equations

    ✔️  How to row reduce a matrix

Here goes WMP! #4... 

Happy solving! Check back on Friday, May 29th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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So, I thought it would be cool to solve this problem two ways--using the elimination method and by using Guass-Jordan elimination on the corresponding augmented matrix--just to compare how they look and see how many steps each requires to solve in completion.

Elimination Method

Guass-Jordan Elimination Method

The elimination method definitely required less steps as compared to the Guass-Jordan elimination method. As you can see, when done correctly, either method will yield the same solution. 👍🏿 The substitution method can also be used. I must say, there was something strangely sort of calming from row reducing the matrix. But then again, I'm not your typical person. 🤷🏿‍♀️ I definitely plan on having a problem from linear algebra for an upcoming WMP!  

 

◾️ Did my solution match yours?? 

◾️ Which method do you prefer when solving a system of four linear equations with four variables??

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

See you soon for WMP! #5 😉😁

Cheers!

The Younge Lady