Question

Solve the following system of equations of elimination by equating the coefficients method: $3x - 4y - 11 = 0,5x - 7y + 4 = 0$.

Answer 1

Hint: When they mention equating the coefficients that means we need to multiply each equation separately by different numbers so that the resultant equations will have common coefficients.

Complete step-by-step answer:
Our system of equations are:
$\begin{gathered}
  3x - 4y - 11 = 0 \\
  5x - 7y + 4 = 0 \\
\end{gathered} $
We can select any variable to be eliminated first and then select the numbers to be multiplied to remove the selected variable terms. So, let us find the value of the numbers first and remove the terms consisting $y$. The coefficient of $y$ in the first equation is (-4) and in the second equation is (-7). Least common multiple of the absolute values of these coefficients = 28. So multiply the first equation by 7 and the second equation by 4. The resultant equations become:
$\begin{gathered}
  21x - 28y = 77 \\
  20x - 28y = - 16 \\
\end{gathered} $
Subtracting second equation from the first equations, we will eliminate y terms,
$\begin{gathered}
   \Rightarrow (21 - 20)x = 77 + 16 \\
   \Rightarrow x = 93 \\
\end{gathered} $
Once we have the value of $x$, we will substitute it for any of the primary equations, to find the value of $y$. We will substitute it in first equation:
$\begin{gathered}
   \Rightarrow 3 \times 93 - 4y = 11 \\
   \Rightarrow 4y = 279 - 11 = 268 \\
   \Rightarrow y = 67 \\
\end{gathered} $
As per suggested method of elimination, we have found the solution, that is $(x,y) = (93,67)$.

Note: This method is closer to the matrix methods applied to simultaneous equations. If it is not mentioned, then you can use any method to solve this method for instance, substitution method.