Question

Check whether the system of equation is inconsistent: $4x+y=23$ and $8x+2y=10$ :
A. Inconsistent
B. Consistent
C. Dependent
D. Non-Linear

Answer 1

Hint:First, we will write down the general form of the linear equations in two variables and then define the conditions for which the given terms will be consistent, inconsistent, and dependent. Then we will find the value of: ${{a}_{1}},{{b}_{1}},{{c}_{1}},{{a}_{2}},{{b}_{2}},{{c}_{2}}$ and check whichever conditions is satisfied. Finally, we will plot the graph in order to prove the answer and hence we will get our answer.

Complete step by step answer:
Let’s understand the definition of the given terms one by one.
A linear system of equation in two variables is generally represented as:
${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\text{ and }{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ ,
Now this system of equations is said to be:
i) Consistent: if it satisfies $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ , here the given lines will intersect only at one point that means it will have only one solution.
ii) Consistent and Dependent: if it satisfies $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}$ , here the lines coincide with one another.
iii) Inconsistent: : If it satisfies $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$ , here both the lines are parallel to one another and have zero solutions.

Now, we have with us: : $4x+y=23$ and $8x+2y=10$ , where:
${{a}_{1}}=4,\text{ }{{b}_{1}}=1,\text{ }{{c}_{1}}=-23,\text{ }{{a}_{2}}=8,\text{ }{{b}_{2}}=2,\text{ }{{c}_{2}}=-5$
Now:
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{4}{8}=\dfrac{1}{2},\text{ }\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{1}{2},\text{ }\dfrac{{{c}_{1}}}{{{c}_{2}}}=\dfrac{-23}{-5}=\dfrac{23}{5}$ which means that $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$ , therefore the given pair of lines are inconsistent.

Hence, the correct answer is option A.
Note: For, these types of questions it is always better to draw graphs as we kind of show proof, like in this question even after our condition is satisfied if we show the graph we will see the lines are parallel which will give an additional point to our answer. Also, be careful while writing down the equations as students can make silly mistakes with the signs in the equations.