Question

How do you solve the following linear system \[2x+6y=5,x-7y=4\]?

Answer 1

Hint: To solve the linear system, we should solve the equation simultaneously. First of all we should add or subtract the equations so that the common term is eliminated. Multiply the first equation by 7 and the second equation by 6. We are doing this to make the variable y equal in both equations, because the variable y in both the equations has different signs and if we add the equation the variable y will be eliminated.

Complete step by step solution:
We have the equations with us, which are \[2x+6y=5,x-7y=4\].
\[\begin{align}
  & 2x+6y=5.....\left( 1 \right) \\
 & x-7y=4....\left( 2 \right) \\
\end{align}\]
Multiply the first equation by 7 and second equation by 6, so that the variable y in both the equations becomes equal. Doing so, we get:
\[\begin{align}
  & 2x+6y=5\left( \times 7 \right) \\
 & x-7y=4\left( \times 6 \right)
\end{align}\]
Now we should multiply the equations. If we multiply them then each variable and the constants will be multiplied. Doing so, we get:
 \[\begin{align}
  & 14x+42y=35 \\
 & 6x-42y=24
\end{align}\]
We should add the equations. If we add the equations then the variable x will be added and the variable y will be eliminated. Variable y will be eliminated because they are the same In both the equations. The constants will also get added. Doing so, we get:
\[\begin{align}
  & \underline{\begin{align}
  & 14x+42y=35 \\
 & 6x-42y=24
\end{align}} \\
 & \,\,\,\,\,20x\,+0y\,=59 \\
\end{align}\]
Hence, the equation after adding them becomes \[20x=59\]. We have to divide by 20 on both the sides such that, we get:
\[\begin{align}
  & \Rightarrow 20x=59 \\
 & \Rightarrow \dfrac{20x}{20}=\dfrac{59}{20} \\
 & \Rightarrow x=\dfrac{59}{20} \\
\end{align}\]
Now we have to solve to find the value of y. To do so we should put \[x=\dfrac{59}{20}\] in \[x-7y=4\], such that we get:
\[\begin{align}
  & \Rightarrow \dfrac{59}{20}-7y=4 \\
 & \Rightarrow - 7y=4-\dfrac{59}{20} \\
 & \Rightarrow - 7y=\dfrac{80-59}{20} \\
 & \Rightarrow - 7y=\dfrac{21}{20} \\
\end{align}\]
Now we have to divide by 7 on both the sides. Such that the coefficient of \[y\] becomes 1.
\[\begin{align}
  & \Rightarrow - 7y=\dfrac{21}{20} \\
 & \Rightarrow \dfrac{7y}{7}=\dfrac{\dfrac{21}{20}}{7} \\
 & \Rightarrow y=\dfrac{-3}{20} \\
\end{align}\]
Hence we have obtained the results of the equation. The values of variable \[x\] and $y$ is \[x=\dfrac{59}{20}\] and \[y=\dfrac{-3}{20}\].

Note: We have obtained the results. If the signs of y in both the equations are the same then we should always keep in mind that we must subtract them and if the signs of y are the same then add the equations. We should never forget to make at least terms equal in both the equations.