An equation that can be put in the form ax + by + c = 0, where a, b and c are real numbers and a, b not equal to zero is called a linear equation in two variables namely x and y. The solution for such an equation is a pair of values, one for x and one for y which further makes the two sides of an equation equal. Let us take an example of linear equation in two variables and understand:

In order to find the solution of Linear equation in two variables, two equations should be known to us.

Consider for Example:

5x + 3y = 30

The above equation has two variables namely x and y.

Graphically this equation can be represented by substituting the variables to zero.

The value of x when y=0 is

5x + 3(0) = 30

\(\Rightarrow x = 6\)

and the value of y when x=0 is,

5(0) + 3y = 30

\(\Rightarrow y = 10\)

Now let us now solve the Linear equation in two variables.

Example: Find the value of variables which satisfies the following equation: 2x + 5y = 20 and 3x+6y =12.

Solution: Using the method of substitution to solve the pair of linear equation, we have:

2x + 5y = 20…………………….(i)

3x+6y =12……………………..(ii)

Multiplying equation (i) by 3 and (ii) by 2, we have:

6x + 15y = 60…………………….(iii)

6x+12y = 24……………………..(iv)

Subtracting equation (iv) from (iii)

3y = 36

\(\Rightarrow y = 12\)

Substituting the value of y in any of the equation (i) or (ii), we have

2x + 5(12) = 20

\(\Rightarrow x = -20\)

Therefore x=-20 and y =12 is the point where the given equations intersect.

In this article we will be considering the situational examples for linear equations in two variables.

Illustration 1:A boat running downstream covers a distance of 20 km in 2 hours while for covering the same distance upstream, it takes 5 hours. What is the speed of the boat in still water?

Solution: These types of questions are the real time example of linear equations in two variables.

In water, the direction along the stream is called downstream . And, the direction against the stream is called upstream .

Let us consider the speed of a boat is u km/h and the speed of the stream is v km/h, then:

Speed Downstream = (u + v) km/h

Speed Upstream = (u – v) km/h

The speed of boat when running downstream = \(\frac{20}{2} km/h\)

The speed of boat when running upstream = \(\frac{20}{5} km/h\)

From above, u + v = 10>…….(1)

u – v = 4 ………. (2)

Adding equation 1 and 2, we get: 2u = 1

u = 7 km/h

Also, v = 3 km/h

Therefore the speed of boat in still water = u = 7 km/h

Illustration 2:A boat running upstream takes 6 hours 30 minutes to cover a certain distance, while it takes 3 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current respectively?

Solution: If the speed downstream is \(a \; km/hr\)

Speed in still water = a + b km/h Rate of stream = \(\frac{1}{2}(a~ -~ b) \)

Let the Boat’s rate upstream be *x* kmph and that downstream be *y* kmph. Then, distance covered upstream in 6 hrs 30 min = Distance covered downstream in 3 hrs.

\(\Rightarrow~x~×~ 6 = y ~×~ 3\)

\(\Rightarrow\frac{13}{2}x = 3y\)

\(\Rightarrow~y = x\)

The required ratio is = \(\frac{y + x}{2}~ :~ \frac{y – x}{2}\)

\(\Rightarrow~\frac{\frac{13x}{6}~+~x}{2}~:~\frac{\frac{13x}{6}~-~x}{2}\)

\(\Rightarrow~\frac{\frac{19x}{6}}{2}~:~\frac{\frac{7x}{6}}{2}\)

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