A set of equations with two variables is called system of linear equations.
A system of linear equations can have
(i) No solution (Parallel lines)
(ii) Infinitely many solutions (Coinciding lines)
(iii) Unique solution (Intersecting lines)
What is solution ?
The point of intersection is known as solution.
No solution |
Infinitely Many |
Intersecting lines |
Parallel will have same slope and different y-intercepts Coinciding lines will have same slope and same y intercept. Intersecting lines will at one point. |
Problem 1 :
x - 3y = 4
2(x - 1) - 6(y + 2) = -6
How many solutions (x, y) are there to the system of equations above ?
(a) zero (b) One (c) Two (d) More than two
Solution :
x - 3y = 4 -----(1)
2(x - 1) - 6(y + 2) = -6 -----(2)
From (2)
2x - 2 - 6y - 12 = -6
2x - 6y = -6 + 12 + 2
2x - 6y = 8
Dividing by 2, we get
x - 3y = 4
Since both are same lines, they will be coinciding lines and it has infinitely many solutions.
Problem 2 :
ax + 4y = 14
5x + 7y = 8
In the system of equations above, a is constant and x and y are variables, If the system has no solution, what is the value of a ?
(a) 20/7 (b) 35/4 (c) -35/4 (d) -20/7
Solution :
ax + 4y = 14
5x + 7y = 18
Since it has no solution, they are parallel lines. So,
m_{1} = m_{2}
ax + 4y = 14 4y = -ax + 14 y = (-a/4) x + (14/4) m_{1 }= -a/4 ----(1) |
5x + 7y = 18 7y = -5x + 18 y = (-5/7)x + 18/7 m_{2 }= -5/7----(2) |
(1) = (2)
-a/4 = -5/7
a = 20/7
Problem 3 :
ax + (1/2)y = 16
4x + 3y = 8
In the system of equations above, a is constant. If the system has no solution, what is the value of a ?
Solution :
ax + (1/2)y = 16
4x + 3y = 8
It has no solutions, they must be parallel and they will have same slopes.
ax + (1/2)y = 16 y/2 = -ax + 16 y = -2ax + 32 m_{1} = -2a |
4x + 3y = 8 3y = -4x + 8 y = (-4/3)x + 8/3 m_{2} = -4/3 |
(1) = (2)
-2a = -4/3
a = 2/3
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