Maharashtra Board Class 9 Maths Part I Chapter 5 Linear Equations in Two Variables PDF Download

Linear Equations in Two Variables

Let's Recall

Ex. Solve the following equations.

(1) m + 3 = 5

m = _____

(2) 3y + 8 = 22

y = _____

(3) \(\frac{x}{3} = 2\)

x = _____

(4) \(2p = p + \frac{4}{9}\)

p = _____

(5) Which number should be added to 5 to obtain 14?

_____ + 5 = 14

x + 5 = 14

x = _____

(6) Which number should be subtracted from 8 to obtain 2?

8 - _____ = 2

8 - y = 2

y = _____

In all above equations, degree of the variable is 1. These are called as Linear equations.

Teacher's Note

When we buy things from a shop, we use simple math like this. If an apple costs 5 rupees and we have 14 rupees, we can find how much money is left.

Exam Trick

Remember: Linear equation means the power of the variable is always 1. Never 2, 3, or higher. Look at the number on top of the variable.

Points to Remember

Linear equation has a variable with power 1 only.

We solve it by moving numbers from one side to other side.

When we move a number, its sign changes: plus becomes minus, and minus becomes plus.

Let's Learn

Linear Equations in Two Variables

Find two numbers whose sum is 14.

Using variables x and y for the two numbers, we can form the equation x + y = 14.

This is an equation in two variables.

We can find many values of x and y satisfying the condition.

e.g. 9 + 5 = 14, 7 + 7 = 14, 8 + 6 = 14, 4 + 10 = 14

(-1) + 15 = 14, 15 + (-1) = 14, 2.6 + 11.4 = 14, 0 + 14 = 14

100 + (-86) = 14, (-100) + (-114) = 14, _____ + _____ = 14, _____ + _____ = 14

Hence above equation has many solutions like (x = 9, y = 5); (x = 7, y = 7); (x = 8, y = 6) etc..

Teacher's Note

Two variables means we have two unknown things. In real life, if we need to find the price of apples and oranges and we know the total cost, we need two equations.

Exam Trick

Remember: Many answers are correct for one equation with two variables. We need two equations to find one single answer.

Points to Remember

Linear equation in two variables has x and y both in it.

One equation with two variables has many solutions.

Each solution is written as an ordered pair like (9, 5) where x = 9 and y = 5.

The order matters: (9, 5) is different from (5, 9).

Conventionally, the Solution

Conventionally, the solution x = 9, y = 5 is written as an ordered pair (9, 5) where 9 is the value of x and 5 is the value of y. To satisfy the equation x + y = 14, we can get infinite ordered pairs like (9, 5), (7, 7), (8, 6), (4, 10), (10, 4), (-1, 15), (2.6, 11.4), ... etc. All of these are the solutions of x + y = 14.

Consider second example.

Find two numbers such that their difference is 2.

Let the greater number be x and the smaller number be y.

Then we get the equation x - y = 2

For the values of x and y, we can get following equations.

10 - 8 = 2, 9 - 7 = 2, 8 - 6 = 2, (-3) - (-5) = 2, 5.3 - 3.3 = 2

15 - 13 = 2, 100 - 98 = 2, _____ - _____ = 2, _____ - _____ = 2

Here if we take values x = 10 and y = 8, then the ordered pair (10, 8) satisfies the above equation. Here we cannot write as (8, 10) because (8, 10) will imply x = 8 and y = 10 and it does not satisfy the equation x - y = 2. Therefore, note that, the order of numbers in the pair indicating solution is very important.

Now let us write the solutions of x - y = 2 in the form of ordered pairs.

(7, 5), (-2, -4), (0, -2), etc. There are infinite solutions.

Find the solution of 4m - 3n = 2.

Construct different equations and find their solutions.

Now observe the first two equations.

x + y = 14 ........ I

x - y = 2 ........ II

Solution of equation I : (9, 5), (7, 7), (8, 6)...

Solutions of Equation II : (7, 5), (-2, -4), (0, -2), (5.2, 3.2), (8, 6)...

(8, 6) is the only common solution of both the equations. This solution satisfies both the equations. Hence it is the unique common solution of both the equations.

When we consider two linear equations in two variables simultaneously and we get unique common solution, then such set of equations is known as Simultaneous equations.

Teacher's Note

When two conditions are given for two unknown things, we can find the exact answer. For example, if we know the total cost of two items and their price difference, we can find each price.

Exam Trick

Remember: Common solution means the same answer works for both equations. Check by putting the answer back in both equations.

Points to Remember

Simultaneous equations are two equations together.

Both equations must have the same two variables x and y.

The common solution satisfies both equations at the same time.

There is only one common solution for two simultaneous equations.

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