# SAT II Math I : Other Mathematical Relationships

## Example Questions

### Example Question #21 : Indirect Proportionality

x varies inversely with y. When x=10, y=6. When x=3, what is y?

Explanation:

Inverse variation takes the form:

Plugging in:

Then solve when x=3:

### Example Question #22 : Indirect Proportionality

The rate of speed is indirectly proportional to the time it takes to travel somewhere. If a person walks at a rate of 3 miles per hour and it takes them 4 hours to get to their destination how far did they travel?

Explanation:

For an indirect proportionality, use the equation

If rate of speed is indirectly proportional to the time, then the speed is y (3 miles per hour) and the time is x (4 hours).

Replace those into the indirect proportionality formula

Solve for k by multiplying by 4 on both sides

The person traveled 12 miles.

### Example Question #41 : Other Mathematical Relationships

Evaluate .

The system has no solution.

The system has no solution.

Explanation:

Rewrite the two equations by setting  and  and substituting:

In terms of  and , this is the system of linear equations:

Solve the first equation for :

Substitute this expression for  in the second equation, then simplify:

The statement is identically false. Therefore, the original system cannot have a solution.

### Example Question #41 : Other Mathematical Relationships

Evaluate .

Explanation:

The system has no solution. The clue can be seen in the second equation

.

The square root of a number (assuming it is real) must be a nonnegative number. Consequently, by the closure of the nonnegative numbers under multiplication, it holds that  and  are nonnegative as well. Also, by the closure of the nonnegative numbers under multiplication,  must also be a positive quantity. Therefore,

cannot have a solution; consequently, neither can the system.

### Example Question #111 : Mathematical Relationships

Evaluate .

No solution

Explanation:

Rewrite the two equations by setting  and  and substituting:

In terms of  and , this is a two-by-two linear system:

Rewrite this as an augmented matrix as follows:

Perform the following row operations to make the left two columns the identity matrix :

and .

Substituting  back for :

Taking the reciprocal of both sides:

.

### Example Question #41 : Other Mathematical Relationships

Evaluate :

No solution

Explanation:

Rewrite the two equations by setting  and  and substituting:

In terms of  and , this is the system of linear equations:

Rewrite this as an augmented matrix as follows:

Perform the following row operations to make the left two columns the identity matrix :

and .

Setting  and solving for  by cubing both sides:

### Example Question #111 : Mathematical Relationships

Which of the following is a cube root of sixty-four?

None of these

None of these

Explanation:

Let  be a cube root of 64. The question is to find a solution of the equation

.

One way to solve this is to subtract 64 from both sides:

64 is a perfect cube, so, as the difference of cubes, the left expression can be factored:

We can set both factors equal to zero and solve:

4 is a cube root of 64; however, this is not one of the choices.

Setting

,

we can make use of the quadratic formula, setting  in the following:

and  are the other two cube roots of ; neither is a choice. Therefore, none of the four choices given are cube roots.