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

## Example Questions

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A company makes toy boats.  Their monthly fixed costs are $1500. The variable costs are$50 per boat.  They sell boats for $75 a piece. How many boats must be sold each month to break even? Possible Answers: 60 75 50 100 Correct answer: 60 Explanation: The break-even point is where the costs equal the revenues Fixed Costs + Variable Costs = Revenues 1500 + 50x = 75x Solving for x results in x = 60 boats sold each month to break even. ### Example Question #1 : How To Find The Solution To An Equation Sally sells cars for a living. She has a monthly salary of$1,000 and a commission of $500 for each car sold. How much money would she make if she sold seven cars in a month? Possible Answers:$4,500

$5,000$4,000

$5,500 Correct answer:$4,500

Explanation:

The commission she gets for selling seven cars is $500 * 7 =$3,500 and added to the salary of $1,000 yields$4,500 for the month.

### Example Question #2 : How To Find The Solution To An Equation

Solve the following system of equations:  xy = 5 and 2x + y = 4.

What is the sum of x and y?

9

1

5

6

1

Explanation:

Add the two equations to get 3x = 9, so x = 3.  Substitute the value of x into one of the equations to find the value of y; therefore x = 3 and y = –2, so their sum is 1.

### Example Question #1 : How To Find The Solution To An Equation

If x = 1/3 and y = 1/2, find the value of 2x + 3y.

13/6

6/5

1

2

13/6

Explanation:

Substitute the values of x and y into the given expression:

2(1/3) + 3(1/2)

= 2/3 + 3/2

= 4/6 + 9/6

= 13/6

### Example Question #2 : How To Find The Solution To An Equation

For what value(s) of  is the expression  undefined?

The expression is undefined for

The expression is undefined for  and

The expression is defined for all real values of

The expression is undefined for  and

The expression is undefined for

The expression is defined for all real values of

Explanation:

The expression is undefined for exactly those values of  which yield a denominator of 0 - that is, for which

However, for all real

,

and, subsequently,

meaning the denominator is always positive. Therefore, the expression is defined for all real values of .

### Example Question #5 : How To Find The Solution To An Equation

Albert has thirteen bills in his wallet, each one a five-dollar bill or a ten-dollar bill. What is the fewest number of ten-dollar bills that he can have and have more than $100. Possible Answers: Correct answer: Explanation: Let be the number of ten-dollar bills Albert has; then he has five-dollar bills. He then has dollars in his wallet, which must be greater than$100. Set up and solve an inequality:

Therefore, the lowest whole number of ten-dollar bills that Albert can have is eight.

### Example Question #6 : How To Find The Solution To An Equation

Solve for :

Explanation:

Expand both products, the left using distribution, the right using the binomial square pattern:

Note that the quadratic terms can be eliminated, yielding a linear equation.

### Example Question #3 : How To Find The Solution To An Equation

Solve for  :

The equation has no solution.

The equation has no solution.

Explanation:

This identically false statement alerts us to the fact that the original equation has no solution.

### Example Question #8 : How To Find The Solution To An Equation

Explanation:

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then collecting all of the terms on the left side:

Use the  method to split the middle term into two terms; we want the coefficients to have a sum of 1 and a product of . These numbers are , so we do the following:

Set each expression equal to 0 and solve:

or

The solution set is .

### Example Question #9 : How To Find The Solution To An Equation

Solve for :

Explanation:

First, rewrite the quadratic equation in standard form by distributing the  through the product on the left, then collecting all of the terms on the left side:

Use the  method to factor the quadratic expression ; we are looking to split the linear term by finding two integers whose sum is 7 and whose product is . These integers are , so:

Set each expression equal to 0 and solve:

or

The solution set is .

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