Question

How‌ ‌do‌ ‌you‌ ‌solve‌ ‌the‌ ‌system‌ ‌\[7x+2y=11\]‌ ‌and‌ ‌\[4x-7y=-10\]‌ ‌by‌ ‌elimination?‌ ‌

Answer 1

Hint: Assume the given equations as equation (1) and (2) respectively. Multiply equation (1) with 7 and equation (2) with 2 and add the two new equations obtained to eliminate the variable y. Solve for the value of x and once the value of x is found, substitute it in equation (2) to solve for the value of y.

Complete step-by-step solution:
Here, we have been provided with the system of equations: \[7x+2y=11\] and \[4x-7y=-10\] and we are asked to solve it. That means we have to find the values of the variables x and y. It is also said that we need to use the elimination method.
Now, here we will eliminate one of the variables and find the value of the other variable. After finding the value of this variable we will substitute it in any of the two given equations and find the value of the eliminated variable.
Let us assume the given equations as equation (1) and (2) respectively, so we have,
\[\Rightarrow 7x+2y=11\] - (1)
\[\Rightarrow 4x-7y=-10\] - (2)
Here, we will eliminate the variable y, so multiplying equation (1) with 7 and equation (2) with 2 and adding the two new equations obtained, we get,
\[\begin{align}
  & \Rightarrow \left( 49x+14y \right)+\left( 8x-14y \right)=77+\left( -20 \right) \\
 & \Rightarrow 57x=77-20 \\
 & \Rightarrow 57x=57 \\
\end{align}\]
Dividing both the sides with 57, we get,
\[\Rightarrow x=1\]
So, we have obtained the value of x, therefore substituting this value in equation (2), we get,
\[\begin{align}
  & \Rightarrow 4\left( 1 \right)-7y=-10 \\
 & \Rightarrow 4-7y=-10 \\
 & \Rightarrow 7y=14 \\
\end{align}\]
Dividing both the sides with 7, we get,
\[\Rightarrow y=2\]
Hence, the solution of the given system of equations is given as: - \[\left( x,y \right)=\left( 1,2 \right)\].

Note: One may note that the value x = 1 and y = 2 satisfies both the equations, hence it proves that our answer is correct. We may also apply the substitution method or the cross – multiplication method to solve the system and check if we are getting the same values or not. They can be used as other proofs. It may also be possible that we can solve these equations graphically. Note that you can also eliminate the variable ‘x’ and find the variable ‘y’ first, the answer will be the same.