Algebraic Concepts

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ISEE Upper Level Quantitative Reasoning › Algebraic Concepts

Questions 1 - 10

1

Multiply:

$15 x ^{2}+31xy -24 y^{2}$

$15 x ^{2}-49xy +24 y^{2}$

$15x^{2} +49xy - 24y^{2}$

$15x^{2} - 24y^{2}$

$15x^{2} -31xy+ 24y^{2}$

Explanation

$= 5x \cdot 3x+5x \cdot 8y - 3y \cdot 3x - 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x+5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x - 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}+40xy - 9xy -24 y^{2}$

$= 15 x ^{2}+31xy -24 y^{2}$

2

Which of the following makes the equation true:

$\frac{x}{5} +9 = 12$

Explanation

To find the answer, we will solve for x. So, we get

$\frac{x}{5} +9 = 12$

$\frac{x}{5} +9 -9 = 12-9$

$\frac{x}{5} + 0 = 3$

$\frac{x}{5} = 3$

$\frac{x}{5} \cdot 5 = 3 \cdot 5$

3

Solve the equation for :

$\frac{e^{3x}}{e^x}=3$

Explanation

First, simplify the equation as much as possible:

$\frac{e^{3x}}{e^x}=3\Rightarrow e^{(3x-x)}=3\Rightarrow e^{2x}=3$

Now we can take the of both sides:

$e^{2x}=3\Rightarrow ln\ e^{2x}=ln\ 3$

$\Rightarrow 2x=ln\ 3$

$\Rightarrow x=\frac{ln\ 3}{2}=\frac{1.098}{2}$

$\Rightarrow x=0.549$

4

Multiply:

$15 x ^{2}+31xy -24 y^{2}$

$15 x ^{2}-49xy +24 y^{2}$

$15x^{2} +49xy - 24y^{2}$

$15x^{2} - 24y^{2}$

$15x^{2} -31xy+ 24y^{2}$

Explanation

$= 5x \cdot 3x+5x \cdot 8y - 3y \cdot 3x - 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x+5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x - 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}+40xy - 9xy -24 y^{2}$

$= 15 x ^{2}+31xy -24 y^{2}$

5

Multiply:

$15 x ^{2}+31xy -24 y^{2}$

$15 x ^{2}-49xy +24 y^{2}$

$15x^{2} +49xy - 24y^{2}$

$15x^{2} - 24y^{2}$

$15x^{2} -31xy+ 24y^{2}$

Explanation

$= 5x \cdot 3x+5x \cdot 8y - 3y \cdot 3x - 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x+5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x - 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}+40xy - 9xy -24 y^{2}$

$= 15 x ^{2}+31xy -24 y^{2}$

6

Solve the equation for :

$\frac{e^{3x}}{e^x}=3$

Explanation

First, simplify the equation as much as possible:

$\frac{e^{3x}}{e^x}=3\Rightarrow e^{(3x-x)}=3\Rightarrow e^{2x}=3$

Now we can take the of both sides:

$e^{2x}=3\Rightarrow ln\ e^{2x}=ln\ 3$

$\Rightarrow 2x=ln\ 3$

$\Rightarrow x=\frac{ln\ 3}{2}=\frac{1.098}{2}$

$\Rightarrow x=0.549$

7

Which of the following makes the equation true:

$\frac{x}{5} +9 = 12$

Explanation

To find the answer, we will solve for x. So, we get

$\frac{x}{5} +9 = 12$

$\frac{x}{5} +9 -9 = 12-9$

$\frac{x}{5} + 0 = 3$

$\frac{x}{5} = 3$

$\frac{x}{5} \cdot 5 = 3 \cdot 5$

8

Multiply:

$15 x ^{2}+31xy -24 y^{2}$

$15 x ^{2}-49xy +24 y^{2}$

$15x^{2} +49xy - 24y^{2}$

$15x^{2} - 24y^{2}$

$15x^{2} -31xy+ 24y^{2}$

Explanation

$= 5x \cdot 3x+5x \cdot 8y - 3y \cdot 3x - 3y \cdot 8y$

$= 5\cdot3 \cdot x \cdot x+5\cdot 8 \cdot x \cdot y - 3\cdot 3\cdot y\cdot x - 3\cdot 8 \cdot y \cdot y$

$= 15 x ^{2}+40xy - 9xy -24 y^{2}$

$= 15 x ^{2}+31xy -24 y^{2}$

9

Solve the equation for :

$\frac{e^{3x}}{e^x}=3$

Explanation

First, simplify the equation as much as possible:

$\frac{e^{3x}}{e^x}=3\Rightarrow e^{(3x-x)}=3\Rightarrow e^{2x}=3$

Now we can take the of both sides:

$e^{2x}=3\Rightarrow ln\ e^{2x}=ln\ 3$

$\Rightarrow 2x=ln\ 3$

$\Rightarrow x=\frac{ln\ 3}{2}=\frac{1.098}{2}$

$\Rightarrow x=0.549$

10

Which of the following makes the equation true:

$\frac{x}{5} +9 = 12$

Explanation

To find the answer, we will solve for x. So, we get

$\frac{x}{5} +9 = 12$

$\frac{x}{5} +9 -9 = 12-9$

$\frac{x}{5} + 0 = 3$

$\frac{x}{5} = 3$

$\frac{x}{5} \cdot 5 = 3 \cdot 5$

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