34.
2x₁ + x₂ = 3
5x₁ + b x₂ = 7.5

The system of linear equations in two variables shown above will have infinite solutions,
if and only if b is equal to __________.

Introduction

The infinite solutions linear equations two variables condition arises when two
linear equations represent the same straight line. This concept is crucial in algebra and
competitive exams. In this article, we determine the value of b for which the
system has infinitely many solutions.

Given System of Equations

2x₁ + x₂ = 3
5x₁ + b x₂ = 7.5

Condition for Infinite Solutions

For a system of two linear equations to have infinite solutions, the ratios of corresponding
coefficients must be equal:

a₁/a₂ = b₁/b₂ = c₁/c₂

Step-by-Step Solution

Compare the ratios:

2/5 = 1/b = 3/7.5

3/7.5 = 0.4
2/5 = 0.4

Therefore,

1/b = 0.4

b = 1 / 0.4 = 2.5

Result

b = 2.5

Substituting b = 2.5, the second equation becomes:

5x₁ + 2.5x₂ = 7.5

This equation is exactly 2.5 times the first equation, proving both represent the same line.

Conditions Explained

• Unique solution: Lines intersect at one point (ratios not equal).
• No solution: Lines are parallel
  (a₁/a₂ = b₁/b₂ ≠ c₁/c₂)
• Infinite solutions: Lines are coincident
  (a₁/a₂ = b₁/b₂ = c₁/c₂)

Verification

With b = 2.5, the equations are dependent and represent the same line. Hence, the system has
infinitely many solutions such as:

x₁ = t,
x₂ = 3 − 2t, where t is any real number.

Practice Table: Testing b Values

Conclusion

The system has infinite solutions only when b = 2.5.
This occurs because both equations become scalar multiples of each other and represent
coincident lines. Always verify coefficient ratios to quickly identify such cases in exams.