# DETERMINE THE NUMBER OF SOLUTIONS TO A SYSTEM OF EQUATIONS

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.

## 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,

m1 = m2

 ax + 4y = 144y = -ax + 14y = (-a/4) x + (14/4)m1 = -a/4 ----(1) 5x + 7y = 187y = -5x + 18y = (-5/7)x + 18/7m2 = -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 = 16y/2 =  -ax + 16y = -2ax + 32m1 = -2a 4x + 3y = 83y = -4x + 8y = (-4/3)x + 8/3m2 = -4/3

(1) = (2)

-2a = -4/3

a = 2/3

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