Maharashtra Board Class 10 Maths Part I Chapter 1 Linear Equations in Two Variables PDF Download

Linear Equations In Two Variables

Let's Study

Methods of solving linear equations in two variables - graphical method, Cramer's method

Equations that can be transformed in linear equation in two variables

Application of simultaneous equations

Let's Recall

Linear Equation In Two Variables

An equation which contains two variables and the degree of each term containing variable is one, is called a linear equation in two variables.

\(ax + by + c = 0\) is the general form of a linear equation in two variables; a, b, c are real numbers and a, b are not equal to zero at the same time.

Example: \(3x - 4y + 12 = 0\) is the general form of equation \(3x = 4y - 12\)

Activity: Complete the following table

Teacher's Note

A linear equation has x and y with power 1 only. For example, in a shop if we buy 2 pencils and 3 notebooks, the cost can be written as 2x + 3y = total cost. This is a linear equation.

Exam Trick

Remember: If x or y has power 2 or higher, or if x and y are multiplied together (like xy), then it is NOT linear. Like \(x^2\) or \(xy\) means not linear.

Points To Remember

Linear equation has two variables only.

The power of each variable is always 1.

General form is ax + by + c = 0.

Both a and b cannot be zero together.

If xy or \(x^2\) or \(y^2\) is there, it is not linear.

Simultaneous Linear Equations

When we think about two linear equations in two variables at the same time, they are called simultaneous equations.

Last year we learnt to solve simultaneous equations by eliminating one variable. Let us revise it.

Example (1): Solve the following simultaneous equations.

\(5x - 3y = 8; 3x + y = 2\)

Solution:

Method I:

\(5x - 3y = 8\) ... (I)

\(3x + y = 2\) ... (II)

Multiplying both sides of equation (II) by 3.

\(9x + 3y = 6\) ... (III)

Now let us add equations (I) and (III)

\(5x - 3y = 8\)

\(9x + 3y = 6\)

\(14x = 14\)

\(\therefore x = 1\)

Substituting x = 1 in equation (II)

\(3x + y = 2\)

\(\therefore 3(1) + y = 2\)

\(\therefore 3 + y = 2\)

\(\therefore y = -1\)

Solution is x = 1, y = -1; it is also written as (x, y) = (1, -1)

Method II:

\(5x - 3y = 8\) ... (I)

\(3x + y = 2\) ... (II)

Let us write value of y in terms of x from equation (II) as

\(y = 2 - 3x\) ... (III)

Substituting this value of y in equation (I).

\(5x - 3y = 8\)

\(\therefore 5x - 3(2 - 3x) = 8\)

\(\therefore 5x - 6 + 9x = 8\)

\(\therefore 14x - 6 = 8\)

\(\therefore 14x = 8 + 6\)

\(\therefore 14x = 14\)

\(\therefore x = 1\)

Substituting x = 1 in equation (III).

\(y = 2 - 3x\)

\(\therefore y = 2 - 3(1)\)

\(\therefore y = 2 - 3\)

\(\therefore y = -1\)

x = 1, y = -1 is the solution.

Teacher's Note

We use elimination method to solve two equations together. For example, if two friends together have 50 rupees and one has 10 rupees more, we use two equations to find how much each has.

Exam Trick

Remember: When you multiply an equation by a number, multiply EVERY term, including the constant on the right side. Don't forget to multiply the right side!

Points To Remember

Simultaneous means both equations at the same time.

We can add or subtract equations to remove one variable.

Always check your answer by putting values back in both equations.

The solution (x, y) must satisfy BOTH equations.

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