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Solving equations reducible to linear form

Introduction:

So far, we've solved a bunch of system of linear equations with two variables. We've used methods like substitution and elimination to solve such equations.
But what happens when the given pair of equations isn't linear?
Well, mostly it gets a lot messier. But sometimes we can convert them to a new set of equations that are linear.
And then try to solve them.
Let's try to understand that by solving one such problem.

The problem:

A boat goes 30km upstream and 44km downstream in 10 hours. It can go 40km upstream and 55km downstream in 13 hours.
The boat has a constant speed in still water and the stream has a constant speed as well.
Moving upstream means moving against the flow and downstream means moving with the flow of water.
Find the speed of the boat in still water.

Strategy:

We are given two scenarios where the boat moves, partly upstream and partly downstream. The distances and total time taken by the boat are known.
What we don't know is the speed of the stream and the speed of the boat in still water. These are our unknown variables.
Using the two scenarios, we can create two equations that involve our two unknowns.
Once we have the equations, we can try to solve them.
If the equations are not linear, we might have to convert them to linear form before solving.

Introducing variables:

Let the speed of the stream be wkmhr.
Let the speed of the boat in still water be bkmhr.
When the boat moves upstream, which is against the flow of water.
Speedup=(bw)kmhr
When the boat moves downstream, which is with the flow of water.
Speeddown=(b+w)kmhr

Framing first equation:

Let's look at the first scenario.
The boat goes 30km upstream and 44km downstream in 10 hours.
We can use the relation between speed, distance and time.
Time=DistanceSpeed
When boat moves upstream,
Timeup=DistanceupSpeedupTimeup=30bwhours
When boat moves downstream,
Timedown=DistancedownSpeeddownTimedown=44b+whours
But the boat took a total time of 10 hours.
Timeup+Timedown=1030bw+44b+w=10
Try to make another equation using the second scenario.

Framing second equation:

Let's look at the second scenario.
The boat goes 40km upstream and 55km downstream in 13 hours.
When boat moves upstream,
Timeup=DistanceupSpeedupTimeup=40bwhours
When boat moves downstream,
Timedown=DistancedownSpeeddownTimedown=55b+whours
But the boat took a total time of 13 hours.
Timeup+Timedown=1340bw+55b+w=13
Now that we have both the equations, let's try to solve them.

Converting to linear form:

Our equations are not linear. Our variables sit in the denominators right now.
But we're in luck as the terms 1bw and 1b+w repeat in both the equations.
30bw+44b+w=1040bw+55b+w=13
If we replace 1bw by x and 1b+w by y, we can get rid of things in the denominator.
30x+44y=1040x+55y=13
Now we're talking! We finally have two linear equations in two variables.
We can proceed from here using either the substitution method or the elimination method.

Solving linear equations for new variables:

30x+44y=1040x+55y=13
We can eliminate x if we multiply the first equation by 4 and second equation by 3, and then subtract one from another.
4×(30x+44y=10)3×(40x+55y=13)
This gives,
120x+176y=40120x+165y=39
Subtracting the second equation from the first,
120x+176y=40(120x+165y=39)11y=1y=111
Plugging the value of y in the first equation:
30x+44×111=1030x+4=1030x=6x=630x=15
Now that we have x and y, we can find b and w.

Solving for initial variables:

Let's plug in the values of x and y in these equations:
1bw=x=151b+w=y=111
This means that:
bw=5b+w=11
Adding both the equations:
2×b=5+112×b=16b=8
Plugging back the value of b:
bw=58w=5w=3
Thus, b or the speed of the boat in still water is 8kmhr.
And w or the speed of the stream is 3kmhr.

Summary:

To solve a pair of equations that are reducible to linear form:
  • Find the expressions that repeat in both the equations. Give them a simpler form: say x and y.
  • Solve the new pair of linear equations for the new variables.
  • Plug back the values of the new variables and solve for the initial variables.
  • Celebrate.

Want to join the conversation?

  • blobby green style avatar for user TRINADH KOLLI
    ritu can row downstream 20km in 2 hours, upstream 4km in 2 hours.find her speed of rowing in still water and speed of current.
    (2 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user tanishqc
      First find the speed of downstream and upstream using "Distance = Speed x Time"

      Then To Calculate Speed Of Rowing And Current,
      Therefore, Speed Of Rowing In Still Water = Speed of Downstream + Speed Of Upstream/2

      For, Speed Of Current = Speed Of Downstream - Speed Of Upstream/2

      Use Get The Required Answer...
      Therefore,
      Speed Of Rowing In Still Water = 6km/hr
      Speed Of Current = 4km/hr
      Please, Verify The Answer As I have Provided U The Correct Answer!!
      (2 votes)