Math Sample Questions

I'll be posting sample math question 1-3 times a week and giving the answer as I post the next problem. Hopefully, we can all find interesting and different ways to solve the problems.

Here is the first question:

1. For what values of x will (x, y) be a solution of the system of equations below:

y = x^2 + 3x -3
2x - 1 = y


A) x = -2 and x = 1
B) x = 0 and x = 1
C) x = -2 and x = -1
D) x = 2 and x = 1
E) x = -2 and x = 3

Source: (removed by moderator)
I am not affiliated to this site so need to attribute for any copyright violation . It offers free questions that one may go through for practice (I like free sites ...

Please keep adding your own questions and lets all attempt to solve this one and others.

 

I have to admit this is one of the more clever SEO social engineering tricks I've seen... if it's legit, I apologize, but we can't permit sigs or URLs to other sites that are basically drive-by spammings.

 

Hey Chip,
Point taken. Won't post any links. Am not linked to the site but I keep using free sites when I tutor online and attribute them to be on the safe side.

But to get back to the core point of the discussion chain, lets see how many of you got the answer right?


There are a number of ways one can solve the set of equations.

Note 1: If there are "n" variables, we need "n" equations to get the solution.

Here we have 2 variables - x and y. And we have 2 equations.


How do you solve an equation with 2 variables?

Step 1: remove 1 variable and make it a 1 variable equation.

y = x^2 + 3x -3
2x - 1 = y

or

x^2 + 3x -3 =y
2x - 1 = y

Subtract the equations and cancel y out

x^2 +x -2 =0

Now we have to solve for the quadratic equation. We can use reverse FOIL method and get

(x+2) (x-1) =0

x +2 =0 => x=-2
x -1 =0 => x= 1

if x =-2, y =2x - 1 = -4-1=-5
if x = 1, y =2x - 1 = 2-1=1

The solution set is (x,y) = (-2,-5) or (1,1)

The question here asks for only "x" so the answer is (A)

Got it?




Also lets move to
Question 2.

Which of the following is the equation of the
line that passes through the points with coordinates
(–2, 1) and (1, 2)?

(A) 2x + y = –3
(B) x + 3y = 7
(C) x + 2y = 0
(D) x – 3y = –5
(E) – x + 2y = 0

Source: CLEP Official Study Guide for College Algebra 18th edition

 

My answer: 42

http://en.wikipedia.org/wiki/42_(number)

 

Can I just say something?? I LOVE how encouraging and refreshing your math posts are! It's awesome Please keep it up!!

 

Thanks cookderosa appreciate the vote of confidence.

Kizmet Great to know you are a voracious reader. Unfortunately, math tutors tend to limit their reading to numbers and not letters and the allure of The Hitchhiker's Guide to the Galaxy doesn't appear to have enticed then yet Alas!!


Q2.
Which of the following is the equation of the
line that passes through the points with coordinates
(–2, 1) and (1, 2)?

(A) 2x + y = –3
(B) x + 3y = 7
(C) x + 2y = 0
(D) x – 3y = –5
(E) – x + 2y = 0


There are 3 ways of solving this.

Method 1.
the equation of the line is
y=mx+b

where m= slope and b = y-intercept.


First, let's find what m is, the slope of the line...

The slope of a line is a measure of how fast the line "goes up" or "goes down". A large slope means the line goes up or down really fast (a very steep line). Small slopes means the line isn't very steep. A slope of zero means the line has no steepness at all; it is perfectly horizontal.
For lines like these, the slope is always defined as "the change in y over the change in x" or, in equation form:

Given that the line passes through the two points P1 = (x1,y1) and P2 = (x2,y2), we first find that the slope of the line is
slope: m = (y2- y1)/ (x2-x1)

So what we need now are the two points you gave that the line passes through. Let's call the first point you gave, (-2,1), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=-2 and y1=1.
Also, let's call the second point you gave, (1,2), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=1 and y2=2.

Now, just plug the numbers into the formula for m above, like this:

m =(2 - 1)/ (1 - -2)

m=1/3
So, we have the first piece to finding the equation of this line, and we can fill it into y=mx+b like this:

y=1/3x+b
Now, what about b, the y-intercept?
To find b, think about what your (x,y) points mean:
(-2,1). When x of the line is -2, y of the line must be 1.
(1,2). When x of the line is 1, y of the line must be 2.
Because you said the line passes through each one of these two points, right?
Now, look at our line's equation so far: y=1/3x+b. b is what we want, the 1/3 is already set and x and y are just two "free variables" sitting there. We can plug anything we want in for x and y here, but we want the equation for the line that specfically passes through the two points (-2,1) and (1,2).

So, why not plug in for x and y from one of our (x,y) points that we know the line passes through? This will allow us to solve for b for the particular line that passes through the two points you gave!.

You can use either (x,y) point you want..the answer will be the same:

(-2,1). y=mx+b or 1=1/3 × -2+b, or solving for b: b=1-(1/3)(-2). b=5/3.
(1,2). y=mx+b or 2=1/3 × 1+b, or solving for b: b=2-(1/3)(1). b=5/3.
See! In both cases we got the same value for b. And this completes our problem.
The equation of the line that passes through the points
(-2,1) and (1,2)

is

y=1/3x+5/3

or 3 y =x +5
or x – 3y = –5

Answer is D

**** Long answer but hope this is helpful****







Method 2

We calculate the slope of the lines formed by two points (–2, 1) and (1, 2)

The slope (as calculated above) = 1/3

We calculate the slope of all the 5 lines given in solution

(A) 2x + y = –3 => y =-2x -3 => slope is -2
(B) x + 3y = 7 => y =-1/3x 7/3 => slope is -1/3
(C) x + 2y = 0 => y = - 1/2x => slope is -1/2
(D) x – 3y = –5 => y =1/3x +5/3 => slope is 1/3
(E) – x + 2y = 0 => y =1/2x => slope is 1/2

Smart idea We know only D and E will have positive slopes (as x and y have opposite signs!). So we need only find slopes of D and E to get the right answer




Method 3.
Plug in the values (–2, 1) and (1, 2) for x and y in each of the 5 options and you will realize that D is the right answer.


Hope this was helpful

Will wait for your comments and post a new question in a few days. If any of you have any math problems you want to solve, POST it here