# Algebra II : Setting Up Equations

## Example Questions

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### Example Question #1 : Setting Up Equations

Tom is painting a fence  feet long. He starts at the West end of the fence and paints at a rate of  feet per hour. After  hours, Huck joins Tom and begins painting from the East end of the fence at a rate of  feet per hour. After  hours of the two boys painting at the same time, Tom leaves Huck to finish the job by himself.

If Huck completes painting the entire fence after Tom leaves, how many more hours will Huck work than Tom?

Explanation:

Tom paints for a total of  hours (2 on his own, 2 with Huck's help). Since he paints at a rate of  feet per hour, use the formula

(or )

to determine the total length of the fence Tom paints.

feet

Subtracting this from the total length of the fence  feet gives the length of the fence Tom will NOT paint:  feet. If Huck finishes the job, he will paint that  feet of the fence. Using , we can determine how long this will take Huck to do:

hours.

If Huck works  hours and Tom works  hours, he works  more hours than Tom.

### Example Question #1 : Equations

If the roots of a function are , what does the function look like in  format?

No equation of this form is possible.

Explanation:

This is a FOIL problem. First, we must set up the problem in a form we can use to create the function. To do this we take the opposite sign of each of the numbers and place them in this format: .

Now we can FOIL.

First:

Outside:

Inside:

Last:

Then add them together to get .

Combine like terms to find the answer, which is .

### Example Question #2 : Equations

Two numbers have a ratio of 5:6 and half of their sum is 22. What are the numbers?

Explanation:

Set up the equation:

Solve the equation:

Find the two numbers:

The two numbers have a ratio of 5:6, therefore the ratio can also be represented as:

The two numbers are 20 and 24.

### Example Question #3 : Equations

Set up an equation that properly displays the information given.

Suzanne has a pack of multi-colored jelly beans. She wants to sort them into equal amounts to give out to her four friends, but not until she eats eight of them. If the total pack contains 60 jelly beans, then how many is each friend going to get?

Explanation:

Let  be the number of jelly beans that each friend will receive. She has four friends, so the total number of jelly beans her friends will receive is . Suzanne eats another eight, so the equation can be written as .

### Example Question #4 : Equations

What is the equation of the line that has a slope of 3 and passes through the point (3,-6)?

Explanation:

The equation for a line in slope-intercept form is:

where  and  are the known coordinates (3,-6).

Substituting gives

and simplifying gives the final answer:

### Example Question #5 : Equations

Your friend goes on a diet to lose a little weight. He starts at  pounds and cuts his calories by  a day. Write a linear equation to express your friend's weight in pounds as a function of weeks on the diet. Hint: there are , calories in a pound.

Explanation:

The question asks for a relation between pounds lost and weeks on the diet. If each day your friend cuts 500 calories, the number of pounds he is losing per week is 1:

The rate of change, or slope, is therefore -1. The slope is negative because the independent variable (weight in pounds) is decreases as the dependent variable (time in weeks) increases. The y-intercept is 180, because that is how much your friend weighs at the start, when time = 0. Plugging these values into  form, we end up with:

### Example Question #1 : Setting Up Equations

A circular tower stands surrounded by a circular moat. A bridge provides a passage over the moat to the tower. The distance from the outer edge of the moat to the center of the tower is  meters. The area of the floor of tower is  . How long is the bridge over the moat?

Explanation:

The distance from the outer edge of the moat to the center of the castle is the radius (100 m) of the larger circle formed by the outer edge of the circular moat.

The radius of the tower's floor (found using the area of the floor), needs to be subracted from 100 m to find the distance fo the bridge.

### Example Question #4 : How To Find Direct Variation

The monthly cost to insure your cars varies directly with the number of cars you own. Right now, you are paying $420 per month to insure 3 cars, but you plan to get 2 more cars, so that you will own 5 cars. How much does it cost to insure 5 cars monthly? Possible Answers: Correct answer: Explanation: The statement, 'The monthly costly to insure your cars varies directly with the number of cars you own' can be mathematically expressed as . M is the monthly cost, C is the number of cars owned, and k is the constant of variation. Given that it costs$420 a month to insure 3 cars, we can find the k-value.

Divide both sides by 3.

Now, we have the mathematical relationship.

Finding how much it costs to insure 5 cars can be found by substituting 5 for C and solving for M.

### Example Question #2 : Setting Up Equations

Express as an equation.

more than  is

Explanation:

Take every word and translate into math.

more than means that you need to add  to something which is .

Anytime you see "is" means equal.

Now let's combine and create an expression of

### Example Question #8 : Equations

Express as an equation.

times  is  less than

Explanation:

Take every word and translate into math.

times something means that you need to multiply  to something which is

less than means that you need to subtract  from .

Anytime you see "is" means equal.

Let's combine to get

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