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!

 Sick for Christmas

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It’s just a few days before my wife, my 16 month-old son and I start our Christmas vacation, and we’re spending it the same way we spent Christmas last year: dealing with a sick baby. For those of you non-parents out there, trust me, this is even worse that you imagine. Sickness isn’t fun for anyone, but dealing with a sick baby is especially bad because he can’t communicate with us how we can make him feel better. All he can do is cry and hope that we can discern from that particular blend of shrieks how we could help. Throw in the fact that since he’s not sleeping we’re not sleeping and you’ve got a recipe that does not equal fun.

But being sick at Christmastime is especially bad. And it’s all about expectations. When you’re feeling bad at a time when you expected to feel good, it’s feels especially awful. The difference between expectation and reality is a pretty accurate measure of happiness. If I went into the Christmas season knowing it’s a time when we commonly get sick and not expecting much out of it, I might not have been so disappointed.

In the same vein, I see many students who sabotage their chances of coming out of the test prep process happy by bringing in unrealistic expectations. I see students who expect 99th percentile scores with a moderate amount of effort even though nothing in their transcript suggests that’s at all realistic. At the end of the process, even if they’ve made great strides and accomplished quite a bit they feel as if they have failed.

So how do you avoid that letdown? First, make your goal to do the best job preparing for the test than you can do. That doesn’t sound as daunting as getting a 2300 or a 165 or a 34 or 720, but it’s an extremely high standard. The expectation is that you will do whatever is in your power to prepare and you will spend your study time as well as you possibly can. You gauge success not by the score that comes in the mail, but by how ready you are when you walk into that test room.

When that test score arrives it tells you which doors are open to you, and which doors may not be, but you don’t run the risk of the bubble popping when you realize your dream doesn’t match reality.

Set reasonable goals for yourself, aim to meet them every day, and stretch the next day’s goal a little further. Realistic expectations allow you to be proud of what you’ve accomplished. Unrealistic goals lead to a feeling almost as bad as being sick at Christmas.

 

 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?