Gruber's Shortest SAT Test Answer Guide
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Gruber's Shortest SAT Test Answer Guide
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In the following questions you must find an answer without referring to choices:

5. If x + y = 7 and xy = 4, then find the value of x2 + y2.

Correct answer:

41: Right!
But don't try to solve for both x and y. You are not asked to do that—you are asked to find the value of x2 + y2 so try to manipulate the equations to get that quantity.
Use Strategy 4, p. 80; BASIC SKILL - Mini Math Refresher - Algebra, p. 156; Math Refresher (409), p. 246: Remember and use classic forms, like
(x + y)2 = x2 + 2xy + y2.

We have (1) (x + y)2 = x2 + 2xy + y2 = 7 x 7 = 49, and since xy = 4, we get 2xy = 8.
Substituting 2xy = 8 in Equation (1), we get (x + y)2 = x2 + 2xy + y2 = x2 + 8 + y2 = 49. Subtracting 8 from both sides of the last equation, we get,
x2 + y2 = 41.

Incorrect.

Did you answer 49? Did you multiply x + y = 7 by itself and get 49, and think that you got x2 + y2 = 49?
Use Strategy 4, p. 80; BASIC SKILL - Mini Math Refresher - Algebra, p. 156; Math Refresher (409), p. 246: Remember and use classic forms, like
(x + y)2 = x2 + 2xy + y2.

Don't try to solve for both x and y. You are not asked to do that—you are asked to find the value of x2 + y2, so try to manipulate the equations to get that quantity.

We have (1) (x + y)2 = x2 + 2xy + y2 = 7 x 7 = 49 and since xy = 4, we get 2xy = 8.
Substituting 2xy = 8 in Equation (1), we get
(x + y)2 = x2 + 2xy + y2 = x2 + 8 + y2 = 49. Subtracting 8 from both sides of the last equation, we get
x2 + y2 = 41.

No. Use Strategy 4, p. 80; BASIC SKILL - Mini Math Refresher - Algebra, p. 156; Math Refresher (409), p. 246: Don't try to solve for both x and y. You are not asked to do that—you are asked to find the value of x2 + y2, so try to manipulate the equations to get that quantity.

We have (1) (x + y)2 = x2 + 2xy + y2 = 7 x 7 = 49 and since xy = 4, we get 2xy = 8.
Substituting 2xy = 8 in Equation (1), we get
(x + y)2 = x2 + 2xy + y2 = x2 + 8 + y2 = 49. Subtracting 8 from both sides of the last equation, we get
x2 + y2 = 41.