BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

equation with two variables

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
User avatar
Newbie | Next Rank: 10 Posts
Posts: 4
Joined: Wed Nov 27, 2013 9:32 am

equation with two variables

by amanjena » Wed Nov 12, 2014 6:58 am
What is the largest possible value of y if (x − 3)^2 + (y + 6)^2 = 100?

a= −2
b= 4
c= 5
d= 8
e= 12

I recognised that for for y to be maximized, the value of (x - 3)^2 must be minimized. Since (x - 3)^2 must be positive, the smallest value it can be is 0.

But then I get stuck at this point y^2 + 12y - 55 = 0
Top
Source: — Problem Solving |
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Wed Nov 12, 2014 7:08 am
amanjena wrote:What is the largest possible value of y if (x − 3)^2 + (y + 6)^2 = 100?

a= −2
b= 4
c= 5
d= 8
e= 12
To MAXIMIZE the value of x, we must MINIMIZE the value of (x − 3)².
Since the square of a value cannot be negative, the smallest possible value of (x − 3)² is 0, when x=3.

Substituting x=3 into (x - 3)² + (y + 6)² = 100, we get:
(3 - 3)² + (y + 6)² = 100
0 + (y + 6)² = 100
(y + 6)² = 100
y + 6 = ±10
y = 4 or y = -16.

Thus, the greatest possible value of y is 4.

The correct answer is B.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
Master | Next Rank: 500 Posts
Posts: 447
Joined: Fri Nov 08, 2013 7:25 am
Thanked: 25 times
Followed by:1 members

by Mathsbuddy » Wed Nov 12, 2014 7:11 am
Rearranged:
(y+6)^2 = 100 - (x-3)^2
The minimum value of (x-3)^2 is 0 (when x = 3)
So maximum value of y comes from:
(y+6)^2 = 100
y+6 = 10 (ignoring y + 6 = -10 as this would yield a lower value)
y = 4

ANSWER B
Top
Master | Next Rank: 500 Posts
Posts: 447
Joined: Fri Nov 08, 2013 7:25 am
Thanked: 25 times
Followed by:1 members

by Mathsbuddy » Wed Nov 12, 2014 7:14 am
amanjena wrote:What is the largest possible value of y if (x − 3)^2 + (y + 6)^2 = 100?

a= −2
b= 4
c= 5
d= 8
e= 12

I recognised that for for y to be maximized, the value of (x - 3)^2 must be minimized. Since (x - 3)^2 must be positive, the smallest value it can be is 0.

But then I get stuck at this point y^2 + 12y - 55 = 0
Hi Amenjena,

Just take a step backwards. No need to expand the brackets and simplify.
(y+6)^2 = 100 is a simple square
so y+6 = +-10
therefore yMAX = 10 - 6 = 4
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Wed Nov 12, 2014 11:38 am
amanjena wrote:What is the largest possible value of y if (x − 3)² + (y + 6)² = 100?

a= −2
b= 4
c= 5
d= 8
e= 12
We can also SOLVE FOR y.
We'll also use the fact that, if y² = k (where k is > 0), then y = +√k or y = -√k

(x − 3)² + (y + 6)² = 100
(y + 6)² = 100 - (x − 3)²
So, applying our green rule, we get: y + 6 = +√[100 - (x − 3)²] or y + 6 = -√[100 - (x − 3)²]
Since our goal is to MAXIMIZE the value of y, we can ignore y + 6 = -√[100 - (x − 3)²]

Let's keep going with y + 6 = +√[100 - (x − 3)²]
Subtract 6 from both sides to get: y = √[100 - (x − 3)²] - 6
So, to MAXIMIZE the value of y, we must MAXIMIZE the value of √[100 - (x − 3)²]
To MAXIMIZE the value of √[100 - (x − 3)²], we must MINIMIZE the value of (x − 3)²
(x − 3)² is minimized when x = 3
So, let's see what y equals when x = 3


We already know that y = √[100 - (x − 3)²] - 6
Plug in to get: y = √[100 - (3 − 3)²] - 6
Simplify: y = √[100 - 0²] - 6
Simplify: y = √[100] - 6
Evaluate: y = 10 - 6 = [spoiler]4 = B[/spoiler]

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
Post new topic Post Reply
• Page 1 of 1