Question

Solve the following pair of linear equations using the elimination method.
$
  4x + 19y + 13 = 0 \\
  13x - 23y + 19 = 0 \\
 $

Answer 1

Hint:-Here we go through by elimination method as in elimination method we have to eliminate one variable by multiplying or dividing the equation to get the coefficient of one variable similar for elimination.

Complete step-by-step answer:
Here in the question the given equations are,
$
  4x + 19y + 13 = 0 \\
  13x - 23y + 19 = 0 \\
 $
Let $4x + 19y + 13 = 0$ it be as equation (1)
And $13x - 23y + 19 = 0$ be as equation (2).
And now try to eliminate the variable ‘x’.
For elimination we have to make the same coefficient of the variables for this we will multiply the equation (1) by 13 and multiply the equation (2) by 4.
Equation (1) multiply by 13 we get,
68x+247y+169 =0 let it be equation (3)
And equation (2) multiply by 4 we get,
68x-92y+76=0 let it be equation (4).
Now subtract the equation (4) from equation (3) we get,
68x+247y+169-(68x-92y+76) =0
68x+247y+169-68x+92y-76=0
339y+93=0
$y = - \dfrac{{93}}{{339}}$
And now put the value of Y in equation (1) we get,
$
  4x + 19 \times \left( { - \dfrac{{93}}{{339}}} \right) + 13 = 0 \\
  4x - \dfrac{{1767}}{{339}} + 13 = 0 \\
  4x = 13 - \dfrac{{1767}}{{339}} \\
  4x = 2640 \\
  \therefore x = 660 \\
 $
Hence the value of x=660 and the value of $y = - \dfrac{{93}}{{339}}$.

Note: - We can also solve this problem by making the coefficient of y same and thus eliminating that first to get the value of $x$ and then substituting this $x$ agai to get the value of $y$, both approaches will give the same answer only.