11.1 Linear Equations II

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.

2.
If the value of one variable in an equation is known, then the value of the other variable can be determined and vice versa.
Example:
Given that 2x + 3y = 6, find the value of
(a) x when y = 4, (b) y when x = –3

Solution:
(a) Substitute y = 4 into the equation.
2x + 3y = 6
2x + 3 (4) = 6
2x + 12 = 6
2= 6 – 12
2= –6
x = –3

(b)   Substitute x = –3 into the equation.
2x + 3y = 6
2 (–3) + 3y = 6
–6 + 3y = 6
3= 6 + 6
3= 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: 2x + y = 9, x = 2+ 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:
Solve the following simultaneous linear equation.
2x + y = 9
3xy = –4

Solution:
(A)  Substitution method
2x + y = 9 -------- (1) label the equations as (1) and (2)
3xy = –4 ------- (2)

From equation (1),
y = 9 – 2x ------- (3) expressing y in terms of x.
Substitute equation (3) into equation (2),
3x – (9 – 2x) = –4
3x – 9 + 2x = –4
5x = –4 + 9
5x = 5
x = 1

Substitute = 1 into equation (1),
2 (1) + y = 9
2 + y = 9
y = 9 – 2
y = 7
The solution is x = 1, y = 7.

(B) Elimination method
2x + y = 9 -------- (1) ← Both equations have the same coefficient of y.
3y = –4 ------- (2)
(1) + (2): 2x + 3x = 9 + (–4) y + (–y) = 0
5x = 5
x = 1

Substitute = 1 into equation (1) or (2),
2x + y = 9 -------- (1)
2 (1) + y = 9
y = 9 – 2
y = 7

The solution is x = 1, y = 7.