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SAT Math : Algebra

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Example Questions

Example Question #22 : How To Simplify An Expression

If x + y = 5 and 3x - y + z = 3 , then 12x + 3z = ?

Possible Answers:

Cannot be determined

Correct answer:

Explanation:

We have three variables and only two equations, so we will not be able to solve for each independent variable. We need to think of another solution.

Notice what happens if we line up the two equations and add them together.

(x + y) + (3x – y + z) = 4x + z

and 5 + 3 = 8

Lets take this equation and multiply the whole thing by 3:

3(4x + z = 8)

Thus, 12x + 3z = 24.

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Example Question #38 : How To Simplify An Expression

$x^{2} = y$

m + 3 = x

3n = m

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

Possible Answers:

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

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

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

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

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

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 #2573 : Sat Mathematics

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

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

Which of the following could be the relationship between and ?

Possible Answers:

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

q = s + 5

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

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

q = s+1

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 #2574 : Sat Mathematics

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

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 : Simplifying Expressions

A = 4x+ 7y

B = 5x- 2y

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

Possible Answers:

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

14x+35

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

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

17x+35

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 #111 : Expressions

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:

7x - 20y

6x-11y

2x + 25y

8x- 28y

x+ 34y

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 #43 : Simplifying Expressions

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:

14x+3y

7x - 20y

13x+12y

2x + 25y

9x+ 5y

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 #801 : Algebra

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

n = -4k

g = k+5

Express in terms of .

Possible Answers:

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

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

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

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

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

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 #802 : Algebra

A = 4x+ 7y

B = 5x- 2y

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

Possible Answers:

3x + 16y

None of the other responses gives a correct answer.

-2x+18y

3x + 12y

-2x+10y

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 #111 : Expressions

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

Possible Answers:

3A-26

57-6A

26-3A

27-3A

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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SAT Math : Algebra

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #22 : How To Simplify An Expression

Example Question #38 : How To Simplify An Expression

Example Question #2573 : Sat Mathematics

Example Question #2574 : Sat Mathematics

Example Question #41 : Simplifying Expressions

Example Question #111 : Expressions

Example Question #43 : Simplifying Expressions

Example Question #801 : Algebra

Example Question #802 : Algebra

Example Question #111 : Expressions

All SAT Math Resources

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