Question

How do you solve the system \[2x+y=0\] and \[3x-y=-10\]?

Answer 1

Hint: In the given question, a pair of linear equations consist of two variables which is ‘x’ and ‘y’. Hence this is the question of linear equations in two variables. A pair of linear equations in two variables can be solved using a substitution method.

Complete step-by-step solution:
We have given the two equations:
\[2x+y=0\]-------- (1)
\[3x-y=-10\]------- (2)
From equation (1), we obtain
\[\Rightarrow y=-2x\]-------- (3)
Substituting the value of \[y=-2x\] in equation (2), we obtain
\[\Rightarrow 3x-y=-10\]
\[\Rightarrow 3x-(-2x)=-10\]
\[\Rightarrow 3x+2x=-10\]
\[\Rightarrow 5x=-10\]
\[\Rightarrow x=\dfrac{-10}{5}=-2\]
\[\Rightarrow x=-2\]
Substituting the value of \[x=-2\] in equation (2), we obtain
\[\Rightarrow 3x-y=-10\]
\[\Rightarrow 3\times (-2)-y=-10\]
\[\Rightarrow -6-y=-10\]
\[\Rightarrow -y=-10+6=-4\]
\[\Rightarrow y=4\]
Therefore, a pair of linear equations in two variables have two solutions.
\[x=-2,\ y=4\]

Additional information:
i) In the given question, no mathematical formula is being used; only the mathematical operations such as addition, subtraction, multiplication and division is used.
ii) While shifting or transferring the numbers to the other side of the equals sign, we use basic mathematical operations and their inverse operations:

Basic operation	Inverse operations
Addition	Subtraction
Subtraction	Addition
Multiplication	Division
Division	Multiplication

Note: A pair of linear equations in two variables have two solutions, one solution for the ‘x’ i.e. the value of ‘x’ and other solution for the ‘y’ i.e. the value of ‘y’. The important thing to recollect about any equation is that the ‘equals’ sign represents a balance. What the sign says is that what’s on the left-hand side is strictly an equal to what’s on the right-hand side. The solution of the equation will not change if the same number is added to or subtracted from both the sides of the equation, multiplying and dividing both the sides of the equations by the same non-zero number.