Question

Which of the following is not a solution of the pair of equations \[3x - 2y = 4\] and \[6x - 4y = 8\] ?
A) $x=2, y=1$
B) $x=4, y=4$
C) $x=6, y=7$
D) $x=5, y=3$

Answer 1

Hint: Here, we are given two equations and we need to find out which of the given options is not a solution. So, for this, we will substitute values of x and y in each equation one by one. Then, the one which does not satisfy the equations, is not the solution of the given equations and thus we will get the required answer.

Complete step by step answer:
Given that, we have two equations: \[3x - 2y = 4\] and \[6x - 4y = 8\]
As we know that, we are given two linear equations in two variables. Also, we can notice that the second equation is 2 times the first equation. Thus, these lines are the concurrent lines means the solution of both of the equations is the same.
We will see who is not a solution of the pairs of these two equations given as below with all the options given.

A) \[x = 2,{\text{ }}y = 1\]
For given two equations:
1) \[3x - 2y = 4\]
Substitute the values of x and y, we will get,
\[ \Rightarrow 3(2) - 2(1) = 4\]
On simplifying this, we will get,
\[ \Rightarrow 6 - 2 = 4\]
\[ \Rightarrow 4 = 4\]

2) \[6x - 4y = 8\]
Substitute the values of x and y, we will get,
\[ \Rightarrow 6(2) - 4(1) = 8\]
On simplifying this, we will get,
\[ \Rightarrow 12 - 4 = 8\]
\[ \Rightarrow 8 = 8\]
Thus, these values of x and y are the solution as it satisfies both the given equations.

B) \[x = 4,{\text{ }}y = 4\]
For given two equations:
1) \[3x - 2y = 4\]
Substitute the values of x and y, we will get,
\[ \Rightarrow 3(4) - 2(4) = 4\]
On simplifying this, we will get,
\[ \Rightarrow 12 - 8 = 4\]
\[ \Rightarrow 4 = 4\]

2) \[6x - 4y = 8\]
Substitute the values of x and y, we will get,
\[ \Rightarrow 6(4) - 4(4) = 8\]
On simplifying this, we will get,
\[ \Rightarrow 24 - 16 = 8\]
\[ \Rightarrow 8 = 8\]
Thus, these values of x and y are the solution as it satisfies both the given equations.

C) \[x = 6,{\text{ }}y = 7\]
For given two equations:
1) \[3x - 2y = 4\]
Substitute the values of x and y, we will get,
\[ \Rightarrow 3(6) - 2(7) = 4\]
On simplifying this, we will get,
\[ \Rightarrow 18 - 14 = 4\]
\[ \Rightarrow 4 = 4\]

2) \[6x - 4y = 8\]
Substitute the values of x and y, we will get,
\[ \Rightarrow 6(6) - 4(7) = 8\]
On simplifying this, we will get,
\[ \Rightarrow 36 - 28 = 8\]
\[ \Rightarrow 8 = 8\]
Thus, these values of x and y are the solution as it satisfies both the given equations.

D) \[x = 5,{\text{ }}y = 3\]
For given two equations:
1) \[3x - 2y = 4\]
Substitute the values of x and y, we will get,
\[ \Rightarrow 3(5) - 2(3) = 4\]
On simplifying this, we will get,
\[ \Rightarrow 15 - 6 = 4\]
\[ \Rightarrow 9 \ne 4\]

2) \[6x - 4y = 8\]
Substitute the values of x and y, we will get,
\[ \Rightarrow 6(5) - 4(3) = 8\]
On simplifying this, we will get,
\[ \Rightarrow 30 - 12 = 8\]
\[ \Rightarrow 18 \ne 8\]
Thus, these values of x and y are not the solution as it does not satisfy both the given equations.

Hence, for the values of \[x = 5\] and\[y = 3\], is not a solution of the pair of the equations \[3x - 2y = 4\;and\;6x - 4y = 8\], as it does not satisfies any of the given equations.

Note:
If we observe the given two equations, the second equation is similar to the first equation. If we multiply the first equation with $2$ on both sides, we will get the second equation. When we plot the graph of these two equations both will be the same. This system of equations has infinitely many solutions.
We can see the graph of the equations along with the given x and y coordinates below: