Simultaneous Equations: Algebra

Happy new year and welcome to Math Sux! In this post we are going to dive right into simultaneous equations and how to solve them three different ways! We will go over how to solve simultaneous equations using the (1) Substitution Method (2) Elimination Method and (3) Graphing Method. Each and every method leading us to the same exact answer! At the end of this post don’t forget to try the practice questions choosing the method that best works for you! Happy calculating! 🙂

What are Simultaneous Equations?

Simultaneous Equations are when two equations are graphed on a coordinate plane and they intersect at, at least one point.  The coordinate point of intersection for both equation is the answer we are trying to find when solving for simultaneous equations. There are three different methods for finding this answer:

We’re going to go over each method for solving simultaneous equations step by step with the example below:

Method #1: Substitution

The idea behind Substitution, is to solve for 1 variable first algebraically, and the plug this value back into the other equation solving for one variable.  Then solving for the remaining variable.  If this sounds confusing, don’t worry! We’re going to do this step by step:

Step 1: Let’s choose the first equation and move our terms around to solve for y.

Simultaneous Equations

Step 3: The equation is set up and ready to solve for x!

Simultaneous Equations

Step 4: All we need to do now, is plug x=3 into one of our original equations to solve for y.

Simultaneous Equations

Step 5: Now that we have solved for both x and y, we have officially found where these two simultaneous equations meet!

Simultaneous Equations

Method #2: Elimination

The main idea of Elimination is to add our two equations together to cancel out one of the variables, allowing us to solve for the remaining variable.  We do this by lining up both equations one on top of the other and adding them together.  If variables at first do not easily cancel out, we then multiply one of the equations by a number so it can. Check out how it’s done step by step below!

Step 1: First, let’s stack both equations one on top of the other to see if we can cancel anything out:

Simultaneous Equations

Step 2: Our goal is to get a 2 in front of y in the first equation, so we are going to multiply the entire first equation by 2.

Simultaneous Equations

Step 3: Now that we multiplied the entire first equation by 2, we can line up our two equations again, adding them together, this time canceling out the variable y to solve for x.

Simultaneous Equations

Step 4: Now, that we’ve found the value of variable x=3, we can plug this into one of our equations and solve for missing unknown variable y.

Simultaneous Equations

Step 5: Now that we have solved for both x and y, we have officially found where these two simultaneous equations meet!

This image has an empty alt attribute; its file name is Screen-Shot-2020-12-30-at-10.21.46-AM.png

Method #3: Graphing

The main idea of Graphing is to graph each a equation on a coordinate plane and then see at what point they intersect.  This is the best method to visualize and check our answer!

Step 1: Before we start graphing let’s convert each equation into y=mx+b (equation of a line) form.

Equation 1:

Equation 2:

Step 2: Now, let’s graph each line, y=3x-4 and y=-x+8, to see at what coordinate point they intersect.

Simultaneous Equations

Need to review how to draw an equation of a line? Check out this post here! Notice we got the same exact answer using all three methods (1) Substitution (2) Elimination and (3) Graphing.

Ready to try the practice problems on your own?! Check them out below!

Practice Questions:

Solve the following simultaneous equations for x and y.

Solutions:

  1. (1, 3)
  2. (4,5)
  3. (-1, -6)
  4. (3, -3)

Want more MathSux?  Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating!

Facebook ~ Twitter ~ TikTok ~ Youtube

TikTok Math Video Compilations

Happy December everyone! With crazy 2020 coming to an end, I thought I would share some TikTok math video compilations of Algebra, Geometry, Algebra 2/Trig, and Statistics for a quick review of all our videos posted throughout the year. Enjoy these TikTok math video compilations and happy calculating! 🙂

Want to make math suck just a little bit less? Subscribe and follow us for FREE fun colorful math videos and lessons every week! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

TikTok Math Video Compilations

Algebra:

Within algebra, you will find arithmetic sequences, combining like terms, box and whisker plots, geometric sequences, solving radical equations, completing the square, 4 ways to factor quadratic equations, piecewise functions and more!

Geometry:

Within Geometry, you will find, how to construct an equilateral triangle, a median of a trapezoid, area of a sector, how to find perpendicular and parallel lines through a given point, SOH CAH TOA right triangle trigonometry, reflections, and more!

Algebra 2/Trig.

Within Algebra 2/Trig., you will find, how to expand a cubed binomial, how to divide polynomials, how to solve log equations, imaginary numbers, synthetic division, unit circle basics, how to graph y=sin(x), and more!

Statistics:

Within statistics, you will find, box and whisker plots, how to find the variance, and, the probability of flipping a coin 2 times!

For full length video, don’t forget to check out our free math video index page! Thanks for stopping by! 🙂

Geometric Sequences: Algebra

Hi everyone and welcome to Mathsux! In this post, we’re going to go over geometric sequences. We’ll see what geometric sequences are, breakdown their formula, and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

What are Geometric Sequences?

Geometric sequences are a sequence of numbers that form a pattern when the same number is either multiplied or divided to each subsequent term.

Example:

geometric sequences

Notice we are multiplying 2 to each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 64. This is a geometric sequence!

In this sequence it’s easy to see what the next term is, but what if we wanted to know the 15th term?  That’s where the Geometric Sequence formula comes in!

Geometric Sequence Formula:

geometric sequences

Now that we broke down our geometric sequence formula, let’s try to answer our original question below:

->First, let’s write out the formula:

geometric sequences

-> Now let’s fill in our formula and solve with the given values.

geometric sequences

Let’s look at another example where, the common ratio is a bit different, and we are dividing the same number from each subsequent term:

-> First let’s identify the common ratio between each number in the sequence. Notice each term in the sequence is being divided by 2 (or multiplied by 1/2 ).

geometric sequences

-> Now let’s write out our formula:

geometric sequences

-> Next let’s fill in our formula and solve with the given values.

Practice Questions:

  1. Find the 12th term given the following sequence: 1250, 625, 312.5, 156.25, 78.125, ….
  2. Find the 17th term given the following sequence: 3, 9, 27, 81, 243,…..
  3. Find the 10th term given the following sequence: 5000, 1250, 312.5, 78.125 …..
  4. Shirley has $100 that she deposits in the bank. She continues to deposit twice the amount of money every month. How much money will she deposit in the twelfth month at the end of the year?

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

*Also, if you want to check out arithmetic sequences click this link here!

Facebook ~ Twitter ~ TikTok ~ Youtube

Arithmetic Sequences: Algebra

Hi everyone and welcome to Mathsux! In this post, we’re going to go over arithmetic sequences. We’ll see what arithmetic sequences are, breakdown their formula, and solve two different types of examples. As always if you want more questions, check out the video below and the practice problems at the end of this post. Happy calculating! 🙂

What are Arithmetic Sequences?

Arithmetic sequences are a sequence of numbers that form a pattern when the same number is either added or subtracted to each subsequent term.

Example:

arithmetic sequences

Notice we are adding 2 to each term in the sequence above. If the pattern were to continue, the next term of the sequence above would be 12. This is an arithmetic sequence!

In the above sequence it’s easy to see what the next term is, but what if we wanted to know the 123rd term?  That’s where the Arithmetic Sequence Formula comes in!

Arithmetic Sequence Formula:

arithmetic sequences

Now that we know the arithmetic sequence formula, let’s try to answer our original question below:

arithmetic sequence examples

-> First, let’s write the arithmetic sequence formula:

arithmetic sequences

-> Fill in our formula and solve with the given values.

Now let’s look at another example where we subtract the same number from each term in the sequence, making the common difference negative.

arithmetic sequence examples

-> First let’s identify the common difference between each number in the sequence. Notice each term in the sequence is being subtracted by 3.

arithmetic sequences

-> Now let’s write out our formula:

arithmetic sequences

-> Next let’s fill in our formula and solve with the given values.

Practice Questions:

  1. Find the 123rd term given the following sequence: 8, 12, 16, 20, 24, ….
  2. Find the 117th term given the following sequence: 2, 2.5, 3, 3.5, …..
  3. Find the 52nd term given the following sequence: 302, 300, 298, …..
  4. A software engineer charges $100 for the first hour of consulting and $50 for each additional hour.  How much would 500 hours of consultation cost?

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Also, if you’re looking to learn more about sequences, check out these posts on Geometric Sequences and Recursive Formulas!

Facebook ~ Twitter ~ TikTok ~ Youtube

Synthetic Division and Factoring Polynomials: Algebra 2/Trig.

Hey there math friends! In this post we will go over how and when to use synthetic division to factor polynomials! So far, in algebra we have gotten used to factoring polynomials with variables raised to the second power, but this post explores how to factor polynomials with variables raised to the third degree and beyond!

If you have any questions don’t hesitate to comment or check out the video below. Also, don’t forget to master your skills with the practice questions at the end of this post. Happy calculating! 🙂

What is Synthetic Division?

Synthetic Division is a shortcut that allows us to easily divide polynomials as opposed to using the long division method. We can only use synthetic division when we divide a polynomial by a binomial in the form of (x-c), where c is a constant number.

Example #1:

*Notice we can use synthetic division in this case because we are dividing by (x+4) which follows our parameters (x-c), where c is equal to 4.

Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division
Synthetic Division

Example #2: Factoring Polynomials

Let’s take a look at the following example and use synthetic division to factor the given polynomial:

Synthetic Division

Check!

The great thing about these questions is that we can always check our work! If we wanted to check our answer, we could simply distribute 2(x+1)(x+3)(x-2) and get our original polynomial, f(x)=2x3+4x2-10x-12.

Try the practice problems on your own below!

Looking to brush up on how to divide polynomials the long way using long division? Check out this post here!

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

How to Construct an Equilateral Triangle?: Geometry

Happy Wednesday math peeps! This post introduces constructions by showing us how to construct an equilateral triangle by using a compass and a ruler. For anyone new to constructions, this is the perfect topic for art aficionados since there is more drawing than there is actual math. 

What is an Equilateral Triangle?

Equilateral Triangle: A triangle with three equal sides.  Not an easy one to forget, the equilateral triangle is super easy to construct given the right tools (compass+ ruler). Take a look below:

how to construct an equilateral triangle

Now, for our Example:

how to construct an equilateral triangle

Solution:

How to Construct an Equilateral Triangle

What’s Happening in this GIF? 

1. Using a compass, measure out the distance of line segment  Screen Shot 2020-08-25 at 4.19.02 PM.

 2. With the compass on point A, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

 3. With the compass on point B, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

4. Notice where the arcs intersect? Using a ruler, connect points A and B to the new point of intersection. This will create two new equal sides of our triangle!

Still got questions? No problem! Don’t hesitate to comment with any questions. Happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

Looking to learn more about triangles? Check out this post on right triangle trigonometry here!

Expanding Cubed Binomials: Algebra 2/Trig.

Greetings math friends! This post will go over expanding cubed binomials using two different methods to get the same answer. We’re so used to seeing squared binomials such as, Screen Shot 2020-08-19 at 11.29.14 AM.png, and expanding them without a second thought.  But what happens when our reliable squared binomials are now raised to the third power, such as,Screen Shot 2020-08-19 at 11.29.48 AM?  Luckily for us, there is a Rule we can use:

Screen Shot 2020-08-18 at 10.12.33 PM

But where did this rule come from?  And how can we so blindly trust it? In this post we will prove why the above rule works for expanding cubed binomials using 2 different methods:

Screen Shot 2020-08-19 at 11.31.13 AM

Why bother? Proving this rule will allow us to expand and simplify any cubic binomial given to us in the future! And since we are proving it 2 different ways, you can choose the method that best works for you.

Method #1: The Box Method

Screen Shot 2020-08-18 at 10.14.37 PM

Step 1: First, focus on the left side of the equation by expanding (a+b)3:

Expanding Cubed Binomials

Step 2: Now we are going to create our first box, multiplying (a+b)(a+b). Notice we put each term of (a+b) on either side of the box. Then multiplied each term where they meet.

Screen Shot 2020-08-18 at 10.15.50 PM

Step 3: Combine like terms ab and ab, then add each term together to get a2+2ab+b2.

Expanding Cubed Binomials

Step 4: Multiply (a2+2ab+b2)(a+b) making a bigger box to include each term.

Expanding Cubed Binomials

Step 5: Now combine like terms (2a2b and a2b) and (2ab2 and ab2), then add each term together and get our answer: a3+3a2b+3ab2+b3.

Expanding Cubed Binomials
Screen Shot 2020-08-18 at 10.21.05 PM.png

Method #2: The Distribution Method

Screen Shot 2020-08-18 at 10.17.54 PM.png

Let’s expand the cubed binomial using the distribution method step by step below:

Expanding Cubed Binomials
Screen Shot 2020-08-18 at 10.21.05 PM.png

Now that we’ve gone over 2 different methods of cubic binomial expansion, try the following practice questions on your own using your favorite method!

Practice Questions: Expand and simplify the following.

Screen Shot 2020-08-18 at 10.21.56 PM

Solutions:

Screen Shot 2020-08-18 at 10.22.19 PM.png

Still, got questions?  No problem! Check out the video above or comment below! Happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

**Bonus: Test your skills with this Regents question on Binomial Cubic Expansion!

How to use Recursive Formulas?: Algebra

Welcome to Mathsux! This post is going to show you everything you need to know about Recursive Formulas by looking at three different examples. Check out the video below for more of an explanation and test your skills with the practice questions at the bottom of this page.  Happy calculating! 🙂

What is a Recursive Formula?

A Recursive Formula is a type of formula that forms a sequence based on the previous term value.  What does that mean?  Check out the example below for a clearer picture:

Example #1:

recursive formula

-> First, let’s decode what these formulas are saying.

recursive formulas
algebra 2 recursive formula
recursive formula examples

-> We found the sequence 2, 6, 10, 14, 18. Since we only needed the fifth term to answer our question, we know our solution is 18.

Example #2:

-> First, let’s decode what these formulas are saying.

algebra 2 recursive formula
algebra 2 recursive formula
algebra 2 recursive formula

***Note this was written in a different notation but is solved in the exact same way!

-> We found the sequence 1,3,9. Since we only needed to find the third term to answer our questions, we know our solution is 9.

-> First, let’s decode what these formulas are saying.

-> We found the sequence 4,10, 28. Since we only needed to find the third term to answer our question, we know the solution is 28.

algebra 2 recursive formula

Practice Questions:

Solutions:

algebra 2 recursive formula

Still got questions? No problem! Check out the video above for more or try the NYS Regents question below, and please don’t hesitate to comment with any questions. Happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

***Bonus! Want to test yourself with a similar NYS Regents question on Recursive Formulas?  Click here!

Intersecting Secants Theorem: Geometry

Ahoy! Today we’re going to cover the Intersecting Secants Theorem!  If you forgot what a secant is in the first place, don’t worry because all it is a line that goes through a circle.  Not so scary right? I was never scared of lines that go through circles before, no reason to start now.

If you have any questions about anything here, don’t hesitate to comment below and check out my video for more of an explanation. Stay positive math peeps and happy calculating! 🙂

Wait, what are Secants?

Screen Shot 2020-07-14 at 10.07.54 PM

Intersecting Secants Theorem: When secants intersect an amazing thing happens! Their line segments are in proportion, meaning we can use something called the Intersecting Secants Theorem to find missing line segments.  Check it out below: 

Intersecting Secants Theorem

Let’s now see how we can apply the intersecting Secants Theorem to find missing length.

Screen Shot 2020-07-14 at 10.45.29 PM.png

Step 1: First, let’s write our formula for Intersecting Secants.

Intersecting Secants Theorem

Step 2: Now fill in our formulas with the given values and simplify.

Intersecting Secants Theorem

Step 3: All we have to do now is solve for x! I use the product.sum method here, but choose the factoring method that best works for you!

Intersecting Secants Theorem

Step 4: Since we have to reject one of our answers, that leaves us with our one and only solution x=2.

Screen Shot 2020-07-14 at 10.14.41 PM.png

Ready to try the practice problems below on your own!?

Practice Questions: Find the value of the missing line segments x.

Intersecting Secants Theorem
Intersecting Secants Theorem

Solutions:

Screen Shot 2020-07-20 at 9.30.55 AM.png

Still got questions?  No problem! Check out the video above or comment below for any questions and follow for the latest MathSux posts. Happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

To review a similar NYS Regents question check out this post here.

Absolute Value Equations: Algebra

Happy Wednesday math friends! Today, we’re going to go over how to solve absolute value equations.  Solving for absolute value equations supplies us with the magic of two potential answers since absolute value is measured by the distance from zero.  And if this sounds confusing, fear not, because everything is explained below!

Also, if you have any questions about anything here, don’t hesitate to comment. Happy calculating! 🙂

Absolute Value measures the “absolute value” or absolute distance from zero.  For example, the absolute value of 4 is 4 and the absolute value of -4 is also 4.  Take a look at the number line below for a clearer picture:

Absolute Value

Now let’s see how we can apply our knowledge of absolute value equations when there is a missing variable!Absolute Value Equations exampleScreen Shot 2020-07-08 at 2.03.46 PM.pngAbsolute Value EquationsScreen Shot 2020-07-08 at 2.04.26 PM.pngAbsolute Value Equations

Screen Shot 2020-07-08 at 2.05.17 PM.png

Absolute Value EquationsNow let’s look at a slightly different example:

Absolute Value Equations exampleScreen Shot 2020-07-08 at 2.07.59 PM

Absolute Value Equations

Screen Shot 2020-07-08 at 2.08.26 PM.png

Absolute Value Equations

Screen Shot 2020-07-08 at 2.09.33 PMAbsolute Value Equations Screen Shot 2020-07-08 at 2.10.39 PM.pngAbsolute Value Equations

Practice Questions: Given the following right triangles, find the missing lengths and side angles rounding to the nearest whole number.

Absolute Value Equations examples

Solutions:

Absolute Value Equations solutions

Still got questions?  No problem! Check out the video the same examples outlined above. Happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

Also, if you’re looking for a review on combining like terms and the distributive property, check out this post here.