Substitute equation 2. into equation 1.,

so, 

Substitute  into equation 2:

so, the solution is .

This problem is a system of equations, and uses the equation .  

Start by assigning variables. Let  stand for the rate of the boat, let  stand for the rate of the current.  

When the boat is going upstream, the total rate is equal to .  You must subtract because the rates are working against each other—the boat is going slower than it would because it has to work against the current.

Using our upstream distance (400m) and time (2hr) from the question, we can set up our rate equation:

When the boat is going downstream, the total rate is equal to  because the boat and current are working with each other, causing the boat to travel faster.

We can refer to the downstream distance (600m) and time (2hr) to set up the second equation:

From here, use elimination to solve for  and .

1. Set up the system of equations, and solve for .

2. Subsitute  into one of the equations to solve for .

Step 1: set up the equation.

Step 2: cross-multiply and rewrite the equation so that  is by itself.

Step 3: Simplify the fraction to find .

Step 1: Multiply each side by the least common multiple (LCM) of the two demonominators. The LCM is twelve.

Step 2: Reduce the fractions.

Step 3: Solve for .

Step 1: Represent the entire right side of the equation as a fraction, using 6 as a common denominator.

Step 2: Cross multiply and solve for .

Step 1: Set up your equation

Consider the rate and driving time of each car.  Additionally, Car 2 leaves an hour after car one, it is necessary to adjust for this in the equation.

Determine the variables:

Let = the rate of Car 1
Let = the rate of Car 2
Let = the amount of time after Car 1 leaves

Therefore:

Step 2: Solve

Solve for x:

Subtract  from each side:

Divide by

Write the second equation in slope-intercept form

This is the same as the first equation.

The two equations represent the same line, therefore there are an infinite number of solutions.

Let  be the number of hours it takes to paint the greenhouse together.

Work problems can be solved by looking at them as rate problems.

Since Tom, Dick, and Harry would take 9, 7, and 11 hours, respectively, to finish the job, the rates at which each works can be viewed as follows:

Tom:  greenhouse per hour;  greenhouse painted.

Dick:   greenhouse per hour;  greenhouse painted.

Harry:   greenhouse per hour;  greenhouse painted.

Since their work adds up to one greenhouse painted, We can set up, and solve for  in, the equation:

Using decimal approximations:

 hours

Since 0.90 hours is approximately equal to  minutes, it will take 2 hours 54 minutes to finish painting the greenhouse - that is, it will be 11:54 AM when they stop. Of the given choices, 12:00 noon comes closest.

Work problems can be solved by looking at them as rate problems.

The hot faucet can fill up the tub at a rate of 16 minutes per tub, or  tub per minute. The cold faucet, similarly, can fill up the tub at a rate of 12 minutes per tub, or  tub per minute. 

Suppose the tub fills up in  minutes. Then, at the end of this time, the hot faucet has filled up  tub, and the cold faucet has filled up  tub, for a total of one tub. We can set up this equation and solve for :

Of the given choices, 7 minutes comes closest.