Simplifying Expressions - GRE Quantitative Reasoning

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) + (3_x –_ y + z) = 4x + z

and 5 + 3 = 8

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

3(4_x_ + z = 8)

Thus, 12_x_ + 3_z_ = 24.

Step 1: 7_x_ + 3_y_

Step 2: 4 * (7_x_ + 3_y_) = 28_x_ + 12_y_

Step 3: 28_x_ + 12_y_ + x = 29_x_ + 12_y_

Step 4: 29_x_ + 12_y_ – (x – y) = 29_x_ + 12_y_ – x + y = 28_x_ + 13_y_

Plugging in numbers is not the best strategy here. Instead, let's see how we can equate the two expressions. Quantity A is actually a difference of squares. 256_x_2 = (16_x_)2 and 49_y_2 = (7_y_)2. These look like the expressions in Quantity B. The formula to remember here is the difference of squares formula, a very important one for this test! a_2 – b_2 = (a + b)(a – b). Thus, if a = 16_x and b = 7_y, 256_x_2 – 49_y_2 = (16_x_ – 7_y_)(16_x_ + 7_y_), and the quantities are equal.

(x + y)2 = x_2 + 2_xy + _y_2

Now, since there are no specifications on what x and y can equal, one or both of them could be 0, making the two columns equal. Any value other than 0 will make the columns unequal because of the additional 2xy term, so the answer cannot be determined.

Quantity A: (x – y)2 = x_2 – 2_xy + _y_2

Quantity B: (x + y)2 = x_2 + 2_xy + _y_2

Both have x_2 + y_2 so cancel those from both columns and just compare –2_xy in Quantity A to 2_xy in Quantity B. If x = 1 and y = 1, –2_xy_ = –2 and 2_xy_ = 2, so Quantity B is greater. But if x = 1 and y = –1, –2_xy_ = 2 and 2_xy_ = –2, so Quantity A is greater. The contradiction means the answer cannot be determined.

If the first integer is n, then n+(n+1)+(n+2)+(n+3)+(n+4)=5n+10

$\frac{5n+10}{5}$=n+2

This is the same as the median.

The expression is simplified as follows:

Since $2^{4}$=16 the value of x must be slightly greater for it to be 17 when raised to the 4th power.

Use substitution to determine the relationship.

For example, we could plug in and .

So far it looks like the first expression is greater, but it's a good idea to try other values of x and y to be sure. This time, we'll try some negative values, say, and .

This time the first quantity is smaller. Therefore the relationship cannot be determined from the information given.

Work from the "inside" outward. Therefore, first solve 4 # 1 by replacing a with 4 and b with 1:

4 # 1 = (4 * 1) + 4 = 4 + 4 = 8

That means: 3 # (4 # 1) = 3 # 8. Solve this now:

3 # 8 = (3 * 8) + 3 = 24 + 3 = 27

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) + (3_x –_ y + z) = 4x + z

and 5 + 3 = 8

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

3(4_x_ + z = 8)

Thus, 12_x_ + 3_z_ = 24.

Step 1: 7_x_ + 3_y_

Step 2: 4 * (7_x_ + 3_y_) = 28_x_ + 12_y_

Step 3: 28_x_ + 12_y_ + x = 29_x_ + 12_y_

Step 4: 29_x_ + 12_y_ – (x – y) = 29_x_ + 12_y_ – x + y = 28_x_ + 13_y_

Plugging in numbers is not the best strategy here. Instead, let's see how we can equate the two expressions. Quantity A is actually a difference of squares. 256_x_2 = (16_x_)2 and 49_y_2 = (7_y_)2. These look like the expressions in Quantity B. The formula to remember here is the difference of squares formula, a very important one for this test! a_2 – b_2 = (a + b)(a – b). Thus, if a = 16_x and b = 7_y, 256_x_2 – 49_y_2 = (16_x_ – 7_y_)(16_x_ + 7_y_), and the quantities are equal.

(x + y)2 = x_2 + 2_xy + _y_2

Now, since there are no specifications on what x and y can equal, one or both of them could be 0, making the two columns equal. Any value other than 0 will make the columns unequal because of the additional 2xy term, so the answer cannot be determined.

Quantity A: (x – y)2 = x_2 – 2_xy + _y_2

Quantity B: (x + y)2 = x_2 + 2_xy + _y_2

Both have x_2 + y_2 so cancel those from both columns and just compare –2_xy in Quantity A to 2_xy in Quantity B. If x = 1 and y = 1, –2_xy_ = –2 and 2_xy_ = 2, so Quantity B is greater. But if x = 1 and y = –1, –2_xy_ = 2 and 2_xy_ = –2, so Quantity A is greater. The contradiction means the answer cannot be determined.

If the first integer is n, then n+(n+1)+(n+2)+(n+3)+(n+4)=5n+10

$\frac{5n+10}{5}$=n+2

This is the same as the median.

The expression is simplified as follows:

Since $2^{4}$=16 the value of x must be slightly greater for it to be 17 when raised to the 4th power.

Use substitution to determine the relationship.

For example, we could plug in and .

So far it looks like the first expression is greater, but it's a good idea to try other values of x and y to be sure. This time, we'll try some negative values, say, and .

This time the first quantity is smaller. Therefore the relationship cannot be determined from the information given.

Work from the "inside" outward. Therefore, first solve 4 # 1 by replacing a with 4 and b with 1:

4 # 1 = (4 * 1) + 4 = 4 + 4 = 8

That means: 3 # (4 # 1) = 3 # 8. Solve this now:

3 # 8 = (3 * 8) + 3 = 24 + 3 = 27

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) + (3_x –_ y + z) = 4x + z

and 5 + 3 = 8

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

3(4_x_ + z = 8)

Thus, 12_x_ + 3_z_ = 24.