Question

The solution of the equation $2x - 3y = 7$ and $4x - 6y = 20$ is
A. $x = 18,y = 12$
B. $x = 0,y = 0$
C. No solution
D. $x = 8,y = 5$

Answer 1

Hint: We can solve the 2 linear equations to get the values of x and y. For solving we can use the method of elimination. If the equation is inconsistent, after addition we get the sum as zero, equal to some constant. If the equation has a solution, we get the values of x and y.

Complete step by step answer:

We have equations
 $2x - 3y = 7$.. (1)
 $4x - 6y = 20$.. (2)
We can solve the equation to find the point of intersection.
We can multiply the 1st equation with 2 and subtract it from equation (2).
 $
  \,\,\,\,\,\,\,\,4x - 6y = 20 \\
  \underline {\left( - \right)4x - 6y = 14} \\
  \,\,\,\,\,\,\,\,0x + 0y = 6 \\
 $
 $ \Rightarrow 0 = 6$
So there is no value of x and y that satisfy the above condition. So the lines do not intersect with each other.
Thus we have no solution of x and y that satisfies both the equation. So, the equation is inconsistent.
Therefore, the correct answer is option C.

Note: We can check the consistency of 2 equations by checking their slope. Graphically the solution of two equations is the point of intersection of the curves formed by the equation. If the two lines intersect on a plane, there will a unique solution. If the lines are parallel, they will never intersect, so it will be inconsistent. If the 2 equations represent the same line, then there will infinite solutions.
An alternate method is to compare the slope of the equations.
For that we need to convert the given equation to the form $y = mx + c$
So the first equation $2x - 3y = 7$ will become,
 $3y = 2x - 7$
 $ \Rightarrow y = \dfrac{2}{3}x - \dfrac{7}{3}$
So, slope is $\dfrac{2}{3}$ .
The second equation becomes,
 $4x - 6y = 20$
 $ \Rightarrow 6y = 4x - 20$
 $ \Rightarrow y = \dfrac{4}{6}y - \dfrac{{20}}{6}$
 $ \Rightarrow y = \dfrac{2}{3}x - \dfrac{{10}}{3}$
So the slope is $\dfrac{2}{3}$ .
As the slope of both the lines is the same, they don’t intersect. So they are inconsistent.