Question

How to determine if x+y =-3 and 2x+y=1 has no solution, one solution or infinite number of solutions and find the solution.

Answer 1

Hint: In the above type of question where we need to check whether there is no solution, one solution of infinite solution we first need to have a basic understanding of what that means, after understanding we are going to solve the simultaneous equations that has been stated in the question and check in which type does it lie and then decide the answer accordingly.

Complete step by step solution:
In the type of question stated above we will first get a basic understanding of what happens when we say that there are no solution to a simultaneous equation, we basically mean to say that we will check whether the slope of the two line are same or not if they are same then there will be no intersection which will result in no solution. If the solution is intersecting at a point then we can say the coordinates where the two lines intersect is the solution to the equation provided. If the y-intercept and slope of both the lines are the same we can conclude that there will be an infinite solution as both the lines are actually the same.
So now to check in which part does our simultaneous equation lie we will convert both the equation in y i.e.
y=-3-x…… (1)
y=1-2x….. (2)
in equation 1 the slope is -1 and in equation 2 slope is -2 as the slope are different we can say that these equations will have a solution so we will equate equation 2 in equation 1 which will give us the value of x i.e.
\[\begin{align}
  & \Rightarrow 1-2x=-3-x \\
 & \Rightarrow 1+3=2x-x \\
 & \Rightarrow 4=x \\
\end{align}\]
Now we will substitute the value of x in equation 1 which will result in value of y which is,
\[\begin{align}
  & \Rightarrow y=-3-4 \\
 & \Rightarrow y=-7 \\
\end{align}\]
So, we can conclude that these simultaneous equations will have one solution where x=4 and y=-7.

Note: In the above type of question we generally forget to check the slope of the equation and blindly solve the question that has been provided to us, to solve the simultaneous equation always check the slope of the equation to solve it more easily and much faster.