EXAMPLES OF SOLVING A PAIR OF LINEAR EQUATIONS WITH GIVEN CONDITION

If a pair of linear equations is having the types of solutions given below, then it has to meet out the corresponding conditions. 

Types of Solution

unique solution

infinitely many solutions

 no solution

Condition

 a1/a2    b1/b2

 a1/a2  =  a1/a = c1/c2

a1/a2  =  a1/a ≠  c1/c2

Example 1 :

For which values of a and b does the following pair of linear equations have an infinite many solution

2x + 3y = 7

(a – b) x + (a + b) y = 3 a + b – 2

Solution :

Condition for having infinitely many solutions

a₁/a₂  =  b₁/b₂  =  c₁/c₂

2 x + 3 y – 7 = 0 --------(1)

(a – b) x + (a + b) y – (3 a + b – 2) = 0 --------(2)

From the above information let us take the values of a₁ , a₂, b₁, b₂, c₁ and c ₂

 a1  =  2               b1  =  3                      c1  =  -7

 a2  =  (a-b)          b2  =  (a + b)             c2  =  – (3a + b – 2)

2/(a – b)  =  3/(a + b)  =  -7/-(3a + b – 2)

 2/(a – b)  =  3/(a + b)  =  7/(3a + b – 2)

2/(a – b)  =  7/(3a + b – 2)

2(3a + b – 2)  =  7(a – b)

6 a + 2 b – 4  =  7 a – 7 b

6 a – 7 a + 2 b + 7 b  =  4

-a + 9 b  =  4  ----- (3)

3(3a + b – 2)  =  7(a + b)

9 a + 3 b – 6  =  7 a + 7 b

9 a – 7 a + 3 b – 7 b  =  6

2 a – 4 b  =  6 ----- (4)

(3) ⋅ 2 + (4)

-2a + 18b  =  8

2a - 4b  =  6

---------------

14b  =  14

b  =  1

By applying the value of b in (3), we get

-a + 9(1)  =  4

-a + 9  =  4

-a  =  4 - 9

a  =  5

Example 2 :

For which value of k will the following pair of linear equations have no solution

3 x + y = 1

(2k – 1) x + (k – 1) y = 2 k + 1

Solution :

3x + y – 1  =  0  --------(1)

(2k – 1)x + (k – 1)y – (2 k + 1) = 0  --------(2)

Condition for having no solution                           

a₁/a ₂ = b₁/b ₂ ≠ c₁/c ₂

From the above information let us take the values of a₁ , a₂, b₁, b₂, c₁ and c ₂

 a₁ = 3                    b₁ = 1                    c₁ = -1

 a₂ = (2 k - 1)         b₂ = (k - 1)              c₂ = – (2 k + 1)

3/(2 k – 1)  =  1/(k – 1)

3 (k  - 1)  =  1 (2 k- 1)

3 k – 3  =  2 k – 1

3 k – 2 k  =  -1 + 3

k = 2

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Quadratic Equation Word Problems Worksheet with Answers

    Oct 07, 22 12:25 PM

    Quadratic Equation Word Problems Worksheet with Answers

    Read More

  2. Problems on Quadratic Equations

    Oct 07, 22 12:20 PM

    Problems on Quadratic Equations

    Read More

  3. SAT Math Practice Worksheets

    Oct 07, 22 09:38 AM

    SAT Math Practice Worksheets - Topic wise worksheet with step by step explanation for each question

    Read More