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General

Easy

Question

# In the following system of linear equations, k is a constant and x and y are variables. For what value of k will the system of equations have infinitely many solution?

kx - 3y = 12

4x - 5y = 20

Hint:

### 1. When Both sides of an equation are equal, the said equation is said to have infinitely many solutions.

i.e. When LHS = RHS, the given equation has infinitely many solutions.

2. For Equations with Infinitely Many solutions:-

(a1/a2) = (b1/b2) = (c1/c2)

## The correct answer is: For k = 2.4, the given set of equations will have Infinitely Many Solutions.

### Step-by-step solution:-

From the given information, we have-

kx - 3y = 12 ….................................................. (Equation i)

4x - 5y = 20 ….................................................. (Equation ii)

Comparing the above equations with standard form of equations i.e. ax + by = c, we get-

a1 = k, b1 = -3, c1 = 12

a2 = 4, b2 = -5, c2 = 20

We know that for Equations with no solution-

(a1/a2) = (b1/b2) = (c1/c2)

i.e. k/4 = -3/-5 = 12/20 .................................................. (Equation iii)

∴ k/4 = -3/-5 ........................................................................... (From Equation iii)

∴ k/4 = 3/5

∴ 5k = 4 * 3 .......................................................................... (Cross multiplying)

∴ 5k = 12

∴ k = 12/5

∴ k = 2.4

Final Answer:-

∴ For k = 2.4, the given set of equations will have Infinitely Many Solutions.

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