Question

How do you solve the system \[9a+2b=10\] and \[9a-2b=10\]?

Answer 1

Hint: Assume the given equations as equation (1) and (2) respectively and use the elimination method to solve the system. To eliminate the variable ‘b’ add the given equations and cancel the variable ‘b’ to find the value of ‘a’. Once the value of ‘a’ is found substitute it in equation (2) to get the value of ‘b’.

Complete step-by-step solution:
Here, we have been provided with the system of equations: \[9a+2b=10\] and \[9a-2b=10\] and we are asked to solve it, that means we have to find the values of the variables a and b.
Now, let us use the elimination method to solve the two equations. Here, we will eliminate one of the variables and find the value of another 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,
\[\Rightarrow 9a+2b=10\] - (1)
\[\Rightarrow 9a-2b=10\] - (2)
Here, we will eliminate the variable b. So, adding the above two equations, we get,
\[\begin{align}
  & \Rightarrow \left( 9a+2b \right)+\left( 9a-2b \right)=10+10 \\
 & \Rightarrow 18a=20 \\
 & \Rightarrow a=\dfrac{20}{18} \\
 & \Rightarrow a=\dfrac{10}{9} \\
\end{align}\]
So, we have obtained the value of a, therefore substituting this value in equation (2), we get,
\[\begin{align}
  & \Rightarrow 9\times \left( \dfrac{10}{9} \right)-2b=10 \\
 & \Rightarrow 10-2b=10 \\
 & \Rightarrow 2b=10-10 \\
 & \Rightarrow 2b=0 \\
 & \Rightarrow b=0 \\
\end{align}\]
Hence, the solution of the given system of equations is given as: - \[\left( a,b \right)=\left( \dfrac{10}{9},0 \right)\].

Note: One may note that we can also apply the substitution method or the cross – multiplication method to solve the above question. But it will be suggested not to use the cross – multiplication method unless and until mentioned in the question. This is because it has a lengthy formula. However, you must remember its formula because sometimes such equations will be there in which if we use the elimination or substitution method then it will make the calculation much more difficult in comparison to the cross - multiplication method. Here, in the above question you can also eliminate the variable ‘a’ by subtracting equation (1) and (2) and find the value of ‘b’ first.