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

## Example Questions

### Example Question #21 : Algebraic Concepts

refers to the least integer greater than or equal to .

and  are integers.

Which is the greater quantity?

(a)

(b)

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

Explanation:

(a) Since  is an integer, .

Since  is an integer, .

(b) By closure,  is an integer, so

.

(a) is the greater quantity.

### Example Question #22 : How To Find The Solution To An Equation

refers to the greatest integer less than or equal to .

and  are integers.

Which is greater?

(a)

(b)

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

(a) Since  is an integer, .

Since  is an integer, .

(b) By closure,  is an integer, so

.

This makes (b) greater.

### Example Question #23 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It cannot be determined from the information given.

(a) and (b) are equal.

Explanation:

Substitute  and, subsequently, :

Factor as , replacing the two question marks with integers whose product is  and whose sum is . These integers are .

Break this up into two equations, replacing  for :

or

This has no solution, since  must be nonnegative.

is the only solution, so (a) and (b) must be equal.

### Example Question #24 : How To Find The Solution To An Equation

Consider the line through points  and .

Which is the greater quantity?

(a) The -coordinate of the -intercept of this line

(b) The -coordinate of the -intercept of this line

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) is greater.

Explanation:

The slope of this line is

.

We will use the point-slope form of the line, with this slope and point :

The -coordinate of the -intercept of this line can be found by substituting  and solving for :

The -coordinate of the -intercept of this line can be found by substituting  and solving for :

This makes (a) the greater quantity.

### Example Question #21 : How To Find The Solution To An Equation

The slope of a line is 2; the line does not pass through the origin.

Which is the greater quantity?

(a) The -coordinate of the -intercept

(b) The -coordinate of the -intercept

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation:

Let  be the - and -intercepts, respectively. We know that the line does not pass through the origin - so .

Then the slope is:

Either  or  can be the greater. For example, if , then , and if , then

### Example Question #21 : Equations

.

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Explanation:

, so substitute and use the power of a power rule.

This makes (a) and (b) equal.

### Example Question #21 : How To Find The Solution To An Equation

Which is the greater quantity?

(a) The slope of the line of the equation

(b) The slope of the line of the equation

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(a) is greater.

Explanation:

Both equations are in slope-intercept form, so compare the coefficients of . The coefficients in (a) and (b) are 5 and 4, respectively, so these are the slopes of the lines. The line in (a) has the greater slope.

### Example Question #22 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

Explanation:

Using two different cases, we show that it is impossible to tell which is greater.

Case 1: . Then , and .

Case 2: . Then , and .

### Example Question #29 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:

To solve the system of equations, add the left and right sides of the equation separately:

Divide:

Substitute to get :

is greater.

### Example Question #30 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater