Question

How do you solve \[x + y = 4\] and \[x - y = 6\] ?

Answer 1

Hint: In this question, we have to solve the given algebraic equations. First we need to express one variable of the first equation in terms of the other variable. Then putting this in the second equation we will get the value of one variable. After that, putting the value of that variable in the first equation we will get the value of the other variable and the required solution.

Complete step-by-step solution:
It is given that, \[x + y = 4\] and \[x - y = 6\] .
We need to solve the two equations.
Now, \[x + y = 4...\left( 1 \right)\]
 \[x - y = 6...\left( 2 \right)\]
From \[\left( 1 \right)\] we get,
 \[ \Rightarrow x = 4 - y\]
Now putting this in equation \[\left( 2 \right)\] we get,
 \[ \Rightarrow 4 - y - y = 6\]
Simplifying we get, \[4 - 2y = 6\]
 \[ \Rightarrow 2y = 4 - 6\]
Let us subtract and divide the term and we get
 \[ \Rightarrow y = \dfrac{{ - 2}}{2} = - 1\]
Putting the value of y in equation \[\left( 1 \right)\] we get,
 \[ \Rightarrow x - 1 = 4\]
 \[ \Rightarrow x = 4 + 1 = 5\]
Therefore we get,
 \[x = 5,y = - 1\]

Hence the solution of the equations \[x + y = 4\] and \[x - y = 6\] is \[x = 5,y = - 1\] .

Note: Two equations with the same variables are called systems of equations and the equations in the system are called simultaneous equations. To solve a system of equations means to find an ordered pair of numbers that satisfies both the equations in the system.
We can solve a system of linear equations by the following methods
Elimination method
Substitution method
Elimination method:
In the elimination method, the object is to make the coefficient of one variable the same in both equations so that one variable can be eliminated either by adding the equations together or by subtracting one from the other.
Substitution method:
In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation.