We want to determine after how many hours the generators will have the same amount of fuel left. Then we want to find which generator runs longer. We will do these things one at a time.
Time to Equal Fuel
Let's begin by using a verbal model to write a system of linear equations. We will use x to represent the number of hours the generator runs and y to represent the fuel left in the generator. The slope of our liner equation will be the fuel usage rate for each generator. The first generator contains 60 gallons of fuel and uses 2.5 gallons of fuel per hour.
y= 60- 2.5 x
The second, more efficient generator holds 40 gallons of fuel and runs at 1.5 gallons per hour.
y= 40- 1.5 x
We can combine these equations into a system.
y= 60- 2.5 x & (I) y= 40- 1.5 x & (II)
Now let's graph each equation and look for a point of intersection, which will be the solution to the system of equations.
The graphs seem to intersect at the point (20,10). Finally, let's check if this point is a solution to the system of equations we created. We can check by substituting 20 for x and 10 for y in each equation and checking to see whether the equations produce true statements.
We found that the solution to the system is (20,10). This means that both generators will have the same amount of fuel, 10 gallons, left after running for 20 hours.
Longer-Running Generator
Next, we want to find out which generator runs longer. Let's take another look at the graph we created.
From the graph, we can see that the first generator runs out of fuel after about 24 hours. The second generator runs out of fuel after about 27 hours. This means that the second generator runs longer.
