Question

How do you solve the system of equations $ 2x+y=3 $ and $ 3x-2y=8 $ ?

Answer 1

Hint: We start solving the problem by considering one of the given equations and then find the function of y in terms of x. We then substitute this obtained function in the second equation and then make the necessary calculations to get the value of x. We then substitute the obtained value of x in one of the given equations and then make the necessary calculations to get the value of y. We then write the obtained values of x and y together as $ \left( x,y \right) $ to get the required solution.

Complete step by step answer:
According to the problem, we are asked to solve the equations $ 2x+y=3 $ and $ 3x-2y=8 $ .
Now, we have the equation $ 2x+y=3 $ .
 $ \Rightarrow y=3-2x $ ---(1). Let us substitute this value of y in the equation $ 3x-2y=8 $ .
So, we have $ 3x-2\left( 3-2x \right)=8 $ .
 $ \Rightarrow 3x-6+4x=8 $ .
\[\Rightarrow 7x-6=8\].
\[\Rightarrow 7x=14\].
\[\Rightarrow x=\dfrac{14}{7}\].
\[\Rightarrow x=2\] ---(2).
Let us substitute equation (2) in equation (1).
So, we have $ y=3-2\left( 2 \right) $ .
 $ \Rightarrow y=3-4 $ .
 $ \Rightarrow y=-1 $ .
So, we have found the solution for the given equations $ 2x+y=3 $ and $ 3x-2y=8 $ as $ \left( 2,-1 \right) $ .
$ \therefore $ The solution for the given equations $ 2x+y=3 $ and $ 3x-2y=8 $ is $ \left( 2,-1 \right) $ .

Note:
Whenever we get this type of problem, we first consider one of the given equations to find the value of y in terms of x which will help us to find the required solution. We can also solve this problem by writing the given linear equations in the matrix form $ AX=B $ and then solving them by using the matrix inversion method or crammer’s rule. Similarly, we can expect problems to solve the set of linear equations $ 5x+8y=80 $ and $ 6x+7y=84 $ using Crammer’s rule.