Linear System Definition

We know that if two expressions are equated, we can call it as an equation. If the power of the variable is one, it is called an linear equation.  If two equations, each containing two same variables (unknowns), are called a system of linear equations.

Eg: 4x + 3y = 12, 6x + 5y = 24.  This can also be called as simultaneous equations, because when we solve these equations, we get one set of values (x, y) in such a way that they satisfy both equations. Hence they got their name.  We can also use the word system as well.  These system of linear equations, which will be representing straight lines can be solved by using (1) substitution method (2) Addition or subtraction method (3) Graphs as well.

Example Problems on Linear System.

Ex 1: Solve: a + 3b = 3 and 4a - 5b = 29 by substitution method.

Solution: Given: a + 3b = 3 ->  (1)

4a - 5b = 29 ->  (2)

Therefore (1) =>  a = 3 ** 3b ->  (3)

Therefore (2) =>  4 (3 ** 3b) ** 5b = 29

=>  12 ** 12b ** 5b = 29

=>  ** 17b = 29 ** 12 = 17.

Therefore b = ** 1.

Therefore (3) =>  a = 3 ** 3 (** 1) = 3 + 3 = 6.

Therefore a = 6, b = ** 1.

Ex 2: Solve: 9x + 4y = 5 and 4x ** 5y = 9 by addition or subtraction method.

Solution: Given: 9x + 4y = 5 ->  (1)

4x ** 5y = 9 ->  (2)

In this method we need to eliminate one variable by making their coefficient same.

(!) xx 5 =>  45x + 20y = 25

(2) xx 4 =>  16x ** 20y = 36

=>  61x = 61 =>  x = 1.

Now by substituting x = 1 in (1), we get: 9 (1) + 4y = 5 =>  4y = 5 ** 9 = ** 4.

=>  y = ** 1.

Therefore The solution is x = 1, y = ** 1.

Ex 3: Solve: x + y = 11 and x ** y = ** 3 using graphs.

Solution: Given: x + y = 11 ->  (1)

x ** y = ** 3 ->  (2)

Here form a table for (1) of (2)

and plot them join the lines.

They will meet at a point is the required answer:

(1) =>  x + y = 11

Linear equationT1

(2) =>  x - y = -3

Linear equationT2

Between, if you have problem on these topics All even Numbers, please browse expert math related websites for more help on cbse solved sample papers for class 12.

From the graph, the solution is x = 4, y = 7.
Practice Problems on Linear System.

1. Solve: a + 5b = 18; 3a + 2b = 41 by substitutions method.

[Ans: a = 13, b = 1]

2. Solve by addition or subtraction method: x + 2y = 11, 2x ** y = 2.

[Ans: x = 3, y = 4]

3. Solve graphically the equations: z ** y = 2 and 2z ** y = 7.

[Ans:  y = 3, z = 5]