 # Linear Equations In Two Variables

## Linear Equations In Two Variables Definition

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 a linear equation in two variables and understand the concept in detail.

### Linear Equations in Two Variables Example

In order to find the solution of Linear equation in 2 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

⇒ x = 6

and the value of y when x = 0 is,

5 (0) + 3y = 30

⇒ y = 10 It is now understood that to solve linear equation in two variables, 2 equations have to be known and then the substitution method can be followed. Let’s understand this with a few example questions.

### Linear Equations in Two Variables Questions

Question: 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

⇒ y = 12

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

2x + 5(12) = 20

⇒ x = −20

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

Now, it is important to know the situational examples which are also known as word problems from linear equations in 2 variables.

### Linear Equations in Two Variables Word Problems

Question 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

We know that Speed = Distance/Time

So, the speed of boat when running downstream = (20⁄2) km/h = 10 km/h

The speed of boat when running upstream = (20⁄5) km/h = 4 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 the boat in still water = u = 7 km/h

Question 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 and the speed upstream is b km/hr, then

Speed in still water = a + b km/h

Rate of stream = ½ (a − b) kmph

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.

⇒ x × 6 = y × 3

⇒ 13/2x = 3y

⇒ y = x

The required ratio is = $\frac{y + x}{2}~ :~ \frac{y – x}{2}$ $\Rightarrow~\frac{\frac{13x}{6}~+~x}{2}~:~\frac{\frac{13x}{6}~-~x}{2}$ $\Rightarrow~\frac{\frac{19x}{6}}{2}~:~\frac{\frac{7x}{6}}{2}$

= 19:7