# ISEE Upper Level Quantitative : How to find the solution to an equation

## Example Questions

### Example Question #81 : Algebraic Concepts

Which is the greater quantity?

(A)

(B)

(A) and (B) are equal

(B) is greater

(A) is greater

It is impossible to determine which is greater from the information given

It is impossible to determine which is greater from the information given

Explanation:

The two equations are actually equivalent, as is proved here:

Therefore, we need only test the first equation. However, it can be shown that it is possible for either of the two to be greater or both to be equal; as can be determined from that third equation  , any two values of  and  that add up to  will solve the system, such as , or .

### Example Question #82 : Algebraic Concepts

Which is the greater quantity?

(A)

(B)

(B) is greater

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

Explanation:

, so , making (A) greater.

### Example Question #83 : Algebraic Concepts

Give the -coordinate of the point on the graph of the equation  that has -coordinate .

No such point exists.

Explanation:

The point  is on the graph of the equation . Finding the -coordinate of this point is the same as evaluating  for . Substitute, and we get:

The -coordinate is therefore .

### Example Question #84 : Algebraic Concepts

Give the -coordinate of the point on the graph of the equation  that has -coordinate 64.

No such point exists.

Explanation:

The point  is on the graph of the equation . Finding the -coordinate of this point is the same as evaluating  for . Substitute, and we get:

### Example Question #85 : Algebraic Concepts

Give the -coordinate of the point on the graph of the equation  that has -coordinate 64.

No such point exists.

No such point exists.

Explanation:

The point  is on the graph of the equation . Finding the -coordinate of this point is the same as evaluating  for . Substitute, and we get:

Since the square root of a number must be positive, there is no solution. Therefore, there is no point on this graph with -coordinate 64.

### Example Question #86 : Algebraic Concepts

Give the -coordinate of the point on the graph of the equation  that has -coordinate .

No such point exists.

No such point exists.

Explanation:

The point  is on the graph of the equation . Finding the -coordinate of this point is the same as evaluating  for . Substitute, and we get:

However, there is no number that can be divided into 3 to yield a quotient of 0, so there is no solution. Therefore, there is no point on this graph with -coordinate .

### Example Question #87 : Algebraic Concepts

What is ?

Explanation:

Substitute  for  in the second equation:

### Example Question #88 : Algebraic Concepts

What is  ?

Explanation:

Solve for  in the top equation:

Substitute  for  in the second equation:

### Example Question #89 : Algebraic Concepts

If , then what is an expression for x in terms of y?

Explanation:

To solve this problem, isolate for x. First, move the y term over to the left side. This gives you . Then, multiply both sides by 4. This gives you . Then, distribute the four to the terms inside the parantheses. This gives you a final answer of .

Evaluate .