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Linear Equations

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CAT 2025 Lesson : Linear Equations - Graph, Dependent & Inconsistent Equations

A linear equation is an equation containing one or more variables and constant(s), where the highest degree of the terms in the equation is

1

.

A simultaneous Linear Equation is an equation containing

2

or more linear equations.
For instance,

x + 4y = 2

and

x + y = 5

are simultaneous linear equations.

To solve for

\bm{n}

number of variables, we need

\bm{n}

number of consistent and independent equations.

These are equations that can be derived or deduced from other equations. They do not provide any new information.

For instance if the

2

equations are

x + 2y = 5

-----Eq(

1

)

2x + 4y = 10

-----Eq(

2

)

then Eq(

1

)

\times 2 =

Eq(

2

). Therefore, solving these will give us infinite values where range is not defined.

Other examples are as follows

6x + 4y = 20

a + 2b + c = 11

2y = 10 - 3x

and

6a + 8b - c = 82

4a + 4b - 3c = 60

These equations are two or more linear equations that have no solutions.

For instance if the

2

equations are

x + 2y = 5

-----Eq(

1

)

2x + 4y = 8

-----Eq(

2

)

Note that Eq(

1

)

\times 2 = 2x + 4y = 10

.
Whereas Eq(

2

) is

2x + 4y = 8

As two values are not possible for a given expression, these equations cannot be solved.

Another example is

2x + 3y = 10

and

4x = 18 - 6y

These are equations that have solutions. All other equations come under this category. To solve for

\bm{n}

variables, we need

\bm{n}

consistent and independent equations. The rest of this lesson covers these equations and methods to solve them.

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