Question

The solution of 8x + 5y= 9, 3x+ 2y =4 lie on x + y = k, then k is:
A. 3
B. 4
C. 5
D. 2

Answer 1

Hint: We have two linear equations given in the question in two variables x and y. since we have two equations therefore we can calculate the value for both x and y. the number of variables whose value has to be calculated should be equal to the number of equations.

Complete step-by-step answer:
The two equations are :
\[
   \Rightarrow 8x{\text{ }} + {\text{ }}5y = {\text{ }}9 \\
   \Rightarrow 3x + {\text{ }}2y{\text{ }} = 4 \;
\]
Now we can find the value of x from any of the equations above in terms of y and then substitute this value of x in the other equation. Hence converting that equation into a linear equation in one variable from the linear equation in two variables. Then the value of y can be calculated by the method of transposition or by subtracting, multiplying, adding or dividing the same terms on both sides of the equation.
Therefore finding the value of x from the equation 3x + 2y= 4 we get
Subtracting 2y from both the sides and dividing by 3 on both the sides of the equation,
 $
   \Rightarrow 3x + 2y = 4 \\
   \Rightarrow 3x + 2y - 2y = 4 - 2y \\
   \Rightarrow 3x + 0 = 4 - 2y \;
   \Rightarrow x = \dfrac{{4 - 2y}}{3} = \dfrac{{2(2 - y)}}{3} \\
 $
Now this value of x can b substituted in the other equation 8x + 5y= 9, we get
 $
   \Rightarrow 8x + 5y = 9 \;
   \Rightarrow 8\left( {\dfrac{{2(2 - y)}}{3}} \right) + 5y = 9 \\
 $
Multiplying by 3 at both side of the equation we get,
 $
   \Rightarrow 3 \times 8\left( {\dfrac{{2(2 - y)}}{3}} \right) + 3 \times 5y = 3 \times 9 \\
   \Rightarrow 16(2 - y) + 15y = 27 \\
   \Rightarrow 32 - 16y + 15y = 27 \;
 $
Transposing the variables at one side and the constants at one side, we get
 $
   \Rightarrow 32 - 27 - y = 0 \\
   \Rightarrow 5 = y \;
 $
Now putting this value of y in the value of x calculated above in the terms of y or substituting the value of y in any of the equation we can get the value of x as,
 $
   \Rightarrow x = \dfrac{{2(2 - y)}}{3} \\
  \therefore y = 5 \\
   \Rightarrow x = \dfrac{{2(2 - 5)}}{3} \\
   \Rightarrow x = \dfrac{{4 - 10}}{3} = \dfrac{{ - 6}}{3} = - 2 \;
 $
Hence the value of x= -2 and y= 5, substituting these values to calculate the value of k
 $
   \Rightarrow x + y = k \\
   \Rightarrow - 2 + 5 = k \\
   \Rightarrow 3 = k \;
 $
Hence the correct option is A.
So, the correct answer is “Option C”.

Note: We observe that in the question above we have been given that points x and y lie on the equation x+y=k, therefore the values of x and y must fulfill this equation and we can calculate the value of k correspondingly.