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

## Example Questions

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Solve for :

Explanation:

### Example Question #2 : Equations

Solve the following equation if :

Explanation:

In order to solve this problem, we must substitute in for .

Based on the order of operations we know that we must first deal with the exponents, so we look at , which is .

Next we multiply.

### Example Question #3 : Equations

Choose the best answer from the four choices given.

If ,

then what must equal?

Explanation:

Notice that is greater than by a factor of 4.

and

Thus, must equal 4 times the other side of the given equation, as well (17).

### Example Question #4 : Equations

Choose the best answer from the four choices given.

Richard reaches into his pocket and determines that he has $3.85 in quarters and nickels. When he pulls out his fistful of change, though, he is dismayed to realize that what he thought were quarters are really pennies and what he thought to be nickels are really dimes. If he has 12 pennies, how much is his change worth? Possible Answers: Correct answer: Explanation: If Richard has 12 pennies, that means that he thought he had 12 quarters ($3.00) and 85 cents worth of nickels (17 nickels). Since what he thought were nickels are really dimes, it means that he has \$1.70 worth of dimes to go along with his 12 cents in pennies.

### Example Question #5 : Equations

Using the information given in each question, compare the quantity in Column A to the quantity in Column B.

Column A          Column B

the slope of        the y-intercept

this equation      of this equation

The quantity in Column A is greater.

The quantity in Column B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

The quantity in Column B is greater.

Explanation:

Divide each of the elements of the equation by 5 to get it into form.

Thus, the slope is 1.8 and the y-intercept is 2, so B is the correct answer.

### Example Question #6 : 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

It is impossible to tell from the information given

Explanation:

Rewrite in standard form:

Use the -method to factor the quadratic expression, splitting the middle term using integers whose product is  and whose sum is . These two numbers are , so the equation becomes:

Set each linear binomial to 0 and solve:

or

Therefore, it is not clear whether  is greater than or less than 3.

### Example Question #7 : Equations

Which is the greater quantity?

(a)

(b)

(b) is greater

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Explanation:

Rewrite in standard form:

Use the -method to factor the quadratic expression, splitting the middle term using integers whose product is  and whose sum is . These integers are , so the equation becomes:

Group and solve:

Set each linear binomial to 0 and solve:

or

In either case, .

### Example Question #8 : Equations

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

It is impossible to tell from the information given

Explanation:

Rewrite in standard form:

Factor the quadratic expression as , replacing the question marks with two integers whose product is  and whose sum is 1. These integers are , so the equation becomes:

Set each linear binomial to 0 and solve:

Therefore, it is not clear whether  is greater than or less than 0.

### Example Question #9 : Equations

Which is the greater quantity?

(a)

(b)

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

Explanation:

Rewrite in standard form:

Factor the quadratic expression as , replacing the question marks with two integers whose product is  and whose sum is 7. These integers are , so rewrite:

Set each linear binomial to 0 and solve:

Both solutions are less than 5.

### Example Question #10 : Equations

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:

Rewrite in standard form:

Factor the quadratic expression as , replacing the question marks with two integers whose product is 50 and whose sum is . These integers are , so rewrite the equation as:

Set each linear binomial to 0 and solve:

It is unclear whether  is equal to or greater than 5.

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