System of Equations Problem

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What is the solution to the system of equations shown below?

6x – 3y = -12

4x + 3y = 2

A) (2,2)

B) (-2,-1)

C) (-1,-1)

D) (-1,2)

E) (1,-2)

In order to solve this problem, we first need to understand what it’s asking. So, what is a “system of equations”? Well, that simply means that we have more than one equation that deals with the same variables. Don’t let that name overwhelm you into thinking it’s more complicated than it really is! In this case, we have to linear equations and solving a system of linear equations simply means finding the point where the two lines intersect. At that point you’ll get an x-value and a y-value that work in both equations.

There are several different ways to solve a system of linear equations and trick is to figure out which one is most efficient for the equations that you’re given. The three methods are:

1. Graphing- If you turn your equations into y=mx+b form you should be able to graph them out and figure out where they intercept. The advantage of this method is that it’s very straight-forward. The disadvantage is that it requires you to be very precise with the lines you draw and can be infeasible when you have an intersection that’s relatively far from the origin. Graphing is typically best used to estimate, or verify a solution.

2. Substitution- This method helps you turn unruly two-variable equations into single-variable equations that can easily be solved. You do this by finding one variable in terms of the other. Let’s see how that works:

Take the first equation.

6x – 3y = -12

Solve for one of the variables.

3y = 6x + 12

y= 2x +4

Then, plug that value into the second equation.

4x + 3(2x+4) = 2

4x+ 6x+ 12 = 2

10x = -10

x = -1

Now that you have the value for x, you can plug that back into either one of your equations to find the value for y.

4(-1) +3y = 2

3y= 6

y= 2

The great part about substitution is that you know it’s going to work. The downside is that there are quite a few steps in the process so it can be somewhat time consuming. That’s where our third method comes in.

3. Combination- This method involves adding or subtracting the two equations with each other with the goal of eliminating one of the variables. When combination can be done without too much manipulation, it’s the easiest and quickest way to solve a system of equations. That’s the case here.

6x – 3y = -12

+ 4x + 3y = 2

________________

10x = -10

As you can see here, when we add the two equations together the y variable drops out and you have the value for x so you can solve from there.

Remember not to worry when you see a system of equations. Choose the best method to solve based on the problem you’re given and you’re all set!

 Imagine This

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Imagine you are given a question that describes the following situation:

There is an isosceles triangle inscribed in a circle with radius = 5. The center of the circle is at (0,0). The triangle is symmetrical about the y-axis.

Do you have a picture in your head about what that would look like? If the answer is no, it could be because you’re unfamiliar with some of the terms that were described. Perhaps you’ve forgotten what an isosceles triangle is (a triangle with two sides of equal length) or perhaps you don’t quite remember which one is the y-axis (it’s the vertical one). If that’s the case, it’s time to go back and review some of the core mathematical terms that you’ll need to know in order to be successful on your test. The Barron’s video course for your test would be a great place to start.

However, even if you got that far, there’s potentially another problem. My guess is that most of you have imagined something like this:

But did you also consider that this green triangle is a possibility as well?

The common advice we give when figures in the coordinate plane is that you should draw a picture. This is sound insofar as your brain can much more easily interpret the graphical information than a set a words given to describe that information. However, draw a picture can get you into trouble when you need to draw the picture or even the pictures.

It’s a word of caution that I hope you’ll remember. When you’re given some graphical information in word form it’s great to translate that into a picture. However, make sure you really take the time to dissect what all of the information means and could mean, so that you aren’t overlooking a part of the solution.

Good luck and happy studying!

 The 12 Days of Christmas

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It’s that time of year when you’re hearing all kinds of holiday music everywhere you go. Maybe the music reminds you of old memories with family. Maybe the music stirs generosity in your heart. Or maybe the music causes you to contemplate mathematical problems.

Okay, so unless you’re me, you problem don’t fall into that last category. But, now that you’re reading I started wondering how many total gifts are given in the 12 Days of Christmas song? If you’re not familiar with it, here’s a link to the lyrics.

The answer to this question isn’t so important to me as the process. How would you go about figuring out a question like that? Would you go gift by gift and count on your fingers and toes? Would you go day by day and sum the 12 totals? Would you be able to divine some other solution?

Since this really isn’t a test-type question that you’re likely to face, this is more an exercise in mathematical thinking. How would you go about it?

Once you’re done figuring that out, take a look at this excellent solution to the problem I proposed.

Isn’t math great?

 Thanksgiving Leftovers Part 2

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Auntie Donna makes a special drink for the Thanksgiving family gathering. All the kids love it. Little do they know that it’s just watered down grocery store punch. She has purchased 9 liters of punch from the store. If she wants to make punch that is 0.6 times as strong as the original, how much drink can she make?

A) 12 liters

B) 15 liters

C) 18 liters

D) 27 liters

E) 54 liters

In order to make punch that is 0.6 times as strong as the original that means that for every 1 liter of mixed drink there are 0.6 liters of punch. Consequently, that means there are 0.4 liters of water in every liter of mixed drink. That’s a ratio of 3 parts punch to 2 parts water. Since Donna purchased 9 liters of punch, she needs to add 6 liters of water for a total of 15 liters of mixed drink.