system of equations by elimination

Question

an accountant rides a bus part of the way to work everyday and walks the rest of the way. The bus averages 40mph, and the accountant walks a speed of 4mph. The distance from home to work is 27 miles, and the total time for the trip is 2hrs. Find out how the accountant walks and how far she rides the bus.

 

How far does he walk?

 

how far does he ride the bus?

Answer 1

1. DEFINE YOUR VARIABLES

 

Usually, these are the unknowns that the question is asking for. You want to know how far he walks and how far he rides the bus. So:

 

x = how many miles he walks

y = how many miles he rides the bus

 

 

 

2. WRITE A SYSTEM OF EQUATIONS

 

This is the part where you turn that word problem into a math problem.

 

First, the total number of miles he travels is 27, so the miles walked plus the miles riding must be 27:

 

x + y = 27

 

 

Second, the total trip time is 2 hours. Since (distance) = (rate)×(time), (miles walked) = (walking speed)×(time walking), so x = 4(time walking). Rearrange that a bit to get x/4 = (time walking). Also, (miles riding the bus) = (bus speed)×(time riding the bus), so y = 40(time riding the bus). Rearrange that a bit to get y/40 = (time riding the bus).

 

We don't know how long he spent walking or riding the bus, but we do know the total time for the trip is 2 hours. So (time walking) + (time riding the bus) = 2, which means:

 

x/4 + y/40 = 2

 

You might want to multiply both sides by 40 to get rid of those fractions:

 

40(x/4) + 40(y/40) = 40(2)

10x + y = 80

 

 

 

3. SOLVE THE SYSTEM OF EQUATIONS

 

You're told to use the elimination method. You've got:

 

x + y = 27

10x + y = 80

 

This is a pretty straightforward elimination, since we actually already changed that second equation to get rid of fractions. Just subtract the first equation from the second equation to eliminate y:

 

   10x + y = 80

-  (x  +  y = 27)

--------------------------

   9x  +  0 = 53

 

Solve for x in that equation:

 

9x + 0 = 53

9x = 53

x = 53/9

 

The accountant walks x = 53/9 ≈ 5.89 miles. Plug x = 53/9 into the first equation from above to solve for y:

 

x + y = 27

53/9 + y = 27

y = 27 - 53/9

y = 243/9 - 53/9

y = 190/9

 

The accountant rides the bus for y = 190/9 ≈ 21.11 miles.

Answer 2

Let x = distance walked

      y = distance traveled by bus

 

So, x + y = 27

 

Distance = (Rate)(Time), so Time = Distance/Rate

 

Time walking + Time in bus = 2

 

x/4 + y/40 = 2

 

Multiply both sides by the LCD, 40, to get 10x + y = 80

 

So, we have the following system of equations:

 

    x + y = 27

10x + y = 80

 

Subtract the equations to obtain -9x = -53

 

x = 5 8/9 miles

y = 21 1/9 miles