Question

Find the value of k for which following system of equations has a unique solution:
x + 2y = 3
5x + ky + 7 = 0

Answer 1

Hint: Here we will proceed by using the condition of unique solution i.e. $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$in the given equations. Then we will compare the ratios of the coefficients of the given equations and get the required value of k.

Complete Step-by-Step solution:
Since the system of equations has a unique solution.
Firstly, when the equation is in form-
$
  {a_1}x + {b_1}y + {c_1} = 0 \\
  {a_2}x + {b_2}y + {c_2} = 0 \\
 $
Now we will compare given system of equations-
x + 2y – 3 = 0………………. (1)
5x + ky + 7 = 0………………. (2)
Here ${a_1} = 1,{b_1} = 2,{c_1} = - 3$
And ${a_2} = 3,{b_2} = k,{c_2} = 7$
We know that the condition for a unique solution is-
$\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
Comparing the ratios of the coefficients of the given equations,
We get-
$ \Rightarrow \dfrac{1}{5} \ne \dfrac{2}{k}$
Cross multiplying,
We get-
$1 \times k \ne 2 \times 5$
Or $k \ne 10$
Therefore, for all real values of k, $\left( {k \ne 10} \right)$given systems of equations have unique solutions.
Or in other words, the value of k is all numbers except k = 10.

Note: While solving this question, we must know that the concept of a consistent system which states that a linear system of equations is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system. Also we can use determinant methods to solve this type of question.