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

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3. Simultaneous Linear Equations

3.1 Types of 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
11.

A simultaneous Linear Equation is an equation containing
22 or more linear equations.
For instance,
x+4y=2x + 4y = 2 and x+y=5x + y = 5 are simultaneous linear equations.

To solve for
n\bm{n} number of variables, we need n\bm{n} number of consistent and independent equations.

3.1.1 Dependent Equations

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

For instance if the
22 equations are
x+2y=5x + 2y = 5 -----Eq(11)
2x+4y=102x + 4y = 10 -----Eq(22)

then Eq(
11) ×2=\times 2 = Eq(22). Therefore, solving these will give us infinite values where range is not defined.

Other examples are as follows
6x+4y=206x + 4y = 20                       a+2b+c=11a + 2b + c = 11
2y=103x2y = 10 - 3x          and      6a+8bc=826a + 8b - c = 82
                                                
4a+4b3c=604a + 4b - 3c = 60

3.1.2 Inconsistent Equations

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

For instance if the
22 equations are

x+2y=5x + 2y = 5 -----Eq(11)
2x+4y=82x + 4y = 8 -----Eq(22)

Note that Eq(
11) ×2=2x+4y=10\times 2 = 2x + 4y = 10.
Whereas Eq(
22) is 2x+4y=82x + 4y = 8

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

Another example is
2x+3y=102x + 3y = 10 and 4x=186y4x = 18 - 6y

3.1.3 Consistent and Independent Equations

These are equations that have solutions. All other equations come under this category. To solve for
n\bm{n} variables, we need n\bm{n} consistent and independent equations. The rest of this lesson covers these equations and methods to solve them.

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