SOLUTION: Susan and Dave take a trip to Danford Island. They walk to the dock at 3 mph then get on the ferry that travels at 6 mph. The total distance is 24 miles and it takes 5 and three q

Algebra ->  Customizable Word Problem Solvers  -> Travel -> SOLUTION: Susan and Dave take a trip to Danford Island. They walk to the dock at 3 mph then get on the ferry that travels at 6 mph. The total distance is 24 miles and it takes 5 and three q      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 100273: Susan and Dave take a trip to Danford Island. They walk to the dock at 3 mph then get on the ferry that travels at 6 mph. The total distance is 24 miles and it takes 5 and three quarters hours to arrive. How far do they travel by ferry?
3x + 6y = 24 miles
x + y = 5.75 hours
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax%2B6%2Ay=24
1%2Ax%2B1%2Ay=5.75

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 3 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 1 is 3, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -3 like this:

1%2A%283%2Ax%2B6%2Ay%29=%2824%29%2A1 Multiply the top equation (both sides) by 1
-3%2A%281%2Ax%2B1%2Ay%29=%285.75%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
3%2Ax%2B6%2Ay=24
-3%2Ax-3%2Ay=-17.25

Notice how 3 and -3 add to zero (ie 3%2B-3=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%283%2Ax-3%2Ax%29%2B%286%2Ay-3%2Ay%29=24-17.25

%283-3%29%2Ax%2B%286-3%29y=24-17.25

cross%283%2B-3%29%2Ax%2B%286-3%29%2Ay=24-17.25 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

3%2Ay=6.75

y=6.75%2F3 Divide both sides by 3 to solve for y



y=2.25 Reduce


Now plug this answer into the top equation 3%2Ax%2B6%2Ay=24 to solve for x

3%2Ax%2B6%282.25%29=24 Plug in y=2.25


3%2Ax%2B13.5=24 Multiply



3%2Ax=24-13.5 Subtract 13.5 from both sides

3%2Ax=10.5 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%2810.5%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=7%2F2 Multiply the terms on the right side


So our answer is

x=7%2F2, y=2.25

which also looks like

(7%2F2, 2.25)

Notice if we graph the equations (if you need help with graphing, check out this solver)

3%2Ax%2B6%2Ay=24
1%2Ax%2B1%2Ay=5.75

we get



graph of 3%2Ax%2B6%2Ay=24 (red) 1%2Ax%2B1%2Ay=5.75 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (7%2F2,2.25). This verifies our answer.