**11.1 Linear Equations II**

**11.1.1 Linear Equations in Two Variables**

**1.**A linear equation in two variables is an equation which contains only linear terms and involves two variables.

**If the value of one variable in an equation is known, then the value of the other variable can be determined and vice versa.**

2.

2.

**Example:**

Given that 2

*x*+ 3*y*= 6, find the value of*x*when

*y*= 4, (b)

*y*when

*x*= –3 (a)

Solution:Solution:

**(a)**Substitute

*y*= 4 into the equation.

2

*x*+ 3*y*= 6 2

*x*+ 3 (4) = 6 2

*x*+ 12 = 6 2

*x*= 6 – 12 2

*x*= –6*x*= –3

**(b)**Substitute

*x*= –3 into the equation.

2

*x*+ 3*y*= 6 2 (–3) + 3

*y*= 6 –6 + 3

*y*= 6 3

*y*= 6 + 6 3

*y*= 12*y*= 4

**3.**A linear equation in two variables has many possible solutions.

**11.1.2**

**Simultaneous Linear Equations in Two Variables**

**1.**Two equations are said to be simultaneous linear equations in two variables if

(a) Both are linear equations in two variables, and

(b) Both involve the same variables.

Example: 2

*x*+*y*= 9,*x*= 2*y*+ 1**2.**The solution of two simultaneous linear equations in two variables is any pair of values (

*x*,

*y*) that satisfied both the equations.

**3.**Simultaneous linear equations in two variables can be solved by the

**substitution method**or the

**elimination method**.

Example:

Example:

Solve the following simultaneous linear equation.

2

*x*+*y*= 93

*x*–*y*= –4

Solution:Solution:

*(A)*

*Substitution method*2

*x*+*y*= 9 -------- (1) ← label the equations as (1) and (2)3

*x*–*y*= –4 ------- (2)From equation (1),

*y*= 9 – 2

*x*------- (3) ← expressing

*y*in terms of

*x*.

Substitute equation (3) into equation (2),

3

*x*– (9 – 2*x*) = –43

*x*– 9 + 2*x*= –45

*x*= –4 + 95

*x*= 5

*x***= 1**

Substitute

*x*= 1 into equation (1),2 (1) +

*y*= 92 +

*y*= 9*y*= 9 – 2

*y***= 7**

The solution is

*x*= 1,*y*= 7.

(B) Elimination method(B) Elimination method

2

*x*+*y*= 9 -------- (1) ← Both equations have the same coefficient of*y.*__3__

*x*–*y*= –4 ------- (2)*x*+ 3

*x*= 9 + (–4) ←

*y*+ (–

*y*) = 0 (1) + (2): 2

5

*x*= 5

*x***= 1**

Substitute

*x*= 1 into equation (1) or (2),2

*x*+*y*= 9 -------- (1)2 (1) +

*y*= 9*y*= 9 – 2

*y***= 7**

The solution is

*x*= 1,*y*= 7.