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SAT Math : Expressions

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Example Question #801 : Algebra

$x^{2} = y$

m + 3 = x

3n = m

Express in terms of . You may assume to be positive.

Possible Answers:

$n = 3 \sqrt{y} + 9$

$n = \frac{ \sqrt{y}-3}{3}$

$n = 3 \sqrt{y} - 1$

$n = \frac{ \sqrt{y}+3}{3}$

$n = 3 \sqrt{y} - 9$

Correct answer:

$n = \frac{ \sqrt{y}-3}{3}$

Explanation:

3n = m

$3n \div 3 = m \div 3$

$n = \frac{m}{3}$

m + 3 = x

m + 3 - 3= x - 3

m = x-3

$x^{2} = y$

$x = \sqrt{y}$ - we throw out the other possible value, $x = -\sqrt{y}$ , since we assume positive.

By substituting:

$n = \frac{m}{3}$

$n = \frac{x-3}{3}$

$n = \frac{ \sqrt{y}-3}{3}$

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Example Question #802 : Algebra

$x^{2} + 6x+ 9 = q$

$x^{2} - 4x + 4 = s$

Which of the following could be the relationship between and ?

Possible Answers:

$q = s + 10 \sqrt{s} + 25$

$q = s + 2 \sqrt{s} + 1$

q = s+1

$q = s + 5 \sqrt{s} + 13$

q = s + 5

Correct answer:

$q = s + 10 \sqrt{s} + 25$

Explanation:

In both expressions, the trinomial in can be seen to be a perfect square trinomial:

$q = x^{2} + 6x+ 9$

$q = x^{2} + 2 \cdot 3 \cdot x + 3^{2}$

$q = (x+3)^{2}$

Taking the square root of both sides of this latter equation:

$\pm \sqrt{q} = x+ 3$

and

$x = - 3 \pm \sqrt{q}$

and

$s = x^{2} - 4x + 4$

$s= x^{2} - 2 \cdot 2 \cdot x + 2^{2}$

$s= (x-2)^{2}$

Taking the square root of both sides of this latter equation:

$\pm \sqrt{s} = x-2$

and $x = 2 \pm \sqrt{s}$

By transitivity,

$- 3 \pm \sqrt{q} = 2 \pm \sqrt{s}$

$- 3 \pm \sqrt{q} + 3 = 2 \pm \sqrt{s} + 3$

$\pm \sqrt{q} = 5 \pm \sqrt{s}$

Square both sides:

$(\pm \sqrt{q} )^{2}=( 5 \pm \sqrt{s})^{2}$

$q= 25 \pm 10 \sqrt{s} + s^{2}$

so one of two things is possible:

$q= s^{2} + 10 \sqrt{s}+ 25$

$q= s^{2} - 10 \sqrt{s}+ 25$

This makes

$q= s^{2} + 10 \sqrt{s}+ 25$

the correct choice.

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Example Question #803 : Algebra

$a^{2}+ b^{2}= 100$

4c= a

d = b - 5

Express in terms of . You may assume that a > 0 and b > 0 .

Possible Answers:

$c=\frac{ \sqrt{- d^{2}-10d + 125}}{4}$

$c=\frac{ \sqrt{ d^{2}-10d + 75}}{4}$

$c=\frac{ \sqrt{ d^{2}-10d -75}}{4}$

$c=\frac{ \sqrt{- d^{2}-10d + 75}}{4}$

$c=\frac{ \sqrt{ d^{2}-10d + 125}}{4}$

Correct answer:

$c=\frac{ \sqrt{- d^{2}-10d + 75}}{4}$

Explanation:

d = b - 5 , so

b = d+5

$a^{2}+ b^{2}= 100$ , so by substitution,

$a^{2}+ (d+5)^{2}= 100$

$a^{2} = 100 - (d+5)^{2}$

$a^{2}= 100 -( d^{2}+ 10d + 25)$

$a^{2}= 100 - d^{2}-10d -25$

$a^{2}= - d^{2}-10d + 75$

Since is presumed positive,

$a = \sqrt{- d^{2}-10d + 75}$

Since 4c = a ,

$c = \frac{a}{4}$

$=\frac{ \sqrt{- d^{2}-10d + 75}}{c}$

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Example Question #41 : How To Simplify An Expression

A = 4x+ 7y

B = 5x- 2y

If x+ 7 = 2y , express A + B in terms of .

Possible Answers:

14x+35

$\frac{23}{2} x+ \frac{35}{2}$

17x+35

$14x+ \frac{35}{2}$

$17x+ \frac{35}{2}$

Correct answer:

$\frac{23}{2} x+ \frac{35}{2}$

Explanation:

A + B

= (4x+7y)+ (5x-2y)

= 4x+ 5x+7y-2y

= 9x+5y

x+ 7 = 2y ; if we multiply both sides by $\frac{5}{2}$ , we get

$\frac{5}{2} \cdot 2y = \frac{5}{2} \cdot \left ( x+ 7 \right )$

$5 y = \frac{5}{2} x+ \frac{35}{2}$

By substitution,

A + B

= 9x+5y

$= 9x+ \frac{5}{2} x+ \frac{35}{2}$

$= \frac{23}{2} x+ \frac{35}{2}$

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Example Question #805 : Algebra

A = 4x+ 7y

B = 5x- 2y

If and are the first and second terms, respectively, of an arithmetic sequence, give the fourth term of the sequence in terms of and .

Possible Answers:

2x + 25y

8x- 28y

6x-11y

x+ 34y

7x - 20y

Correct answer:

7x - 20y

Explanation:

The common difference of the sequence can be found by subtracting the first term from the second:

d = B - A = (5x-2y) - (4x+7y) = 5x-4x -2y -7y = x - 9y

Add twice this to the second term to get the fourth term:

B + 2d

$= \left (5x- 2y \right )+ 2 \left ( x - 9y \right )$

= 5x- 2y + 2 x - 18y

= 5x + 2 x - 2y- 18y

= 7x - 20y

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Example Question #806 : Algebra

A = 4x+ 7y

B = 5x- 2y

Now define a sequence as follows:

$a_{1} = A$

$a_{2} = B$

For all n > 2 , $a_{n} = a_{n-1}+ a_{n-2}$ .

Give the fourth term of the sequence.

Possible Answers:

2x + 25y

9x+ 5y

14x+3y

7x - 20y

13x+12y

Correct answer:

14x+3y

Explanation:

Each term, beginning with the third, is the sum of the previous two terms. Therefore, the third term is:

$a_{3} = a_{1}+ a_{2}$

= A+ B

$=\left ( 4x+ 7y \right ) + \left ( 5x- 2y \right )$

= 4x+ 5x + 7y - 2y

= 9x+ 5y

$a_{4} = a_{2}+ a_{3}$

= B+ (9x+5y)

= ( 5x- 2y )+ (9x+5y)

= 5x+ 9x - 2y +5y

=14x+3y

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Example Question #807 : Algebra

$r = \frac{1}{g^{3}}$

n = -4k

g = k+5

Express in terms of .

Possible Answers:

$n=20-4 \sqrt [3] {\frac{1}{r}}$

$n=-20+4 r^{3}$

$n=20+4 \sqrt [3] {r}$

$n=20-4 \sqrt [3] {r}$

$n=20+4 \sqrt [3] {\frac{1}{r}}$

Correct answer:

$n=20-4 \sqrt [3] {\frac{1}{r}}$

Explanation:

g = k+5

g -5= k+5 -5

g-5 = k

$r = \frac{1}{g^{3}}$

$g^{3}= \frac{1}{r}$

$g = \sqrt [3] {\frac{1}{r}}$

Substituting twice:

n = -4k

= -4(g-5)

= -4g + 20

= 20- 4g

$=20-4 \sqrt [3] {\frac{1}{r}}$

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Example Question #41 : Simplifying Expressions

A = 4x+ 7y

B = 5x- 2y

Which of the following expressions is equivalent to 2A - B ?

Possible Answers:

-2x+10y

3x + 12y

3x + 16y

-2x+18y

None of the other responses gives a correct answer.

Correct answer:

3x + 16y

Explanation:

Since we are given and , our first step is to multiply the quanity of A by two. After multiplying A by two, then subtract B from that quantity.

2A - B

$= 2 \left ( 4x+ 7y \right ) - \left ( 5x- 2y \right )$

Remember to distribute the negative sign that is in front of the second binomial to each term within the parentheses.

$= 2 \left ( 4x \right )+ 2 \left (7y \right ) - 5x + 2y$

= 8x + 14 y - 5x + 2y

From here, combine like terms to completely simplify the expression.

= 8x- 5x + 14 y + 2y

= 3x + 16y

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Example Question #809 : Algebra

If 3(2A+B)+4-(B-1)=57 , which expression is equal to?

Possible Answers:

57-6A

26-3A

27-3A

3A-26

Correct answer:

26-3A

Explanation:

The purpose of this question is to practice using the correct order of operations, which is the proper way to simplify this expression. Using PEMDAS (parentheses, exponent, multiplication, division, addition, subtraction) the parentheses are simplified first, making the expression into

6A+3B+4-B+1=57 .

Then the addition and subraction are utilized, leaving

6A+2B+5=57 .

After that, the '6A' and '5' terms will be moved to the other side of the equation to isolate 2B,

2B=52-6A .

Then dividing by two to isolate the B, it leaves the answer as

B=26-3A .

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Example Question #41 : Simplifying Expressions

A utility company charges "F" dollars in fees every month, just for having an account. On top of this, they charge "a" dollars per kgal for the first 100 kgal of water and "b" dollars per kgal for each kgal over 100 . If Joaquin uses "g" kgal of water (where g > 100 ), how much should he expect to pay on his water bill?

Possible Answers:

(a+b)g + F

a(g-100) + b(g) + F

a(g)+b(g+100)+F

a(100)+b(g-100)+F

a(100)+ b(F+g)

Correct answer:

a(100)+b(g-100)+F

Explanation:

Joaquin pays a dollars per kgal for the first 100 kgal only, so a(100) should be a term in our final answer.

Since he pays b dollars per kgal only for the kgal AFTER the first 100, he gets charged for g - 100 kgal at rate b. This means b(g-100) will also be in our final expression.

Finally, we simply add on the F dollars in fees, no need to multiply this by anything since it is a flat charge each month.

a(100)+b(g-100)+F

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SAT Math : Expressions

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #801 : Algebra

Example Question #802 : Algebra

Example Question #803 : Algebra

Example Question #41 : How To Simplify An Expression

Example Question #805 : Algebra

Example Question #806 : Algebra

Example Question #807 : Algebra

Example Question #41 : Simplifying Expressions

Example Question #809 : Algebra

Example Question #41 : Simplifying Expressions

All SAT Math Resources

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