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Question: Solve \(x - 2y = - 6\;{\text{and}}\; - x + y = 2.\)...

Solve x−2y=−6  and  −x+y=2.x - 2y = - 6\;{\text{and}}\; - x + y = 2.

Explanation

Solution

So we have been given a pair of linear equations which can be solved either by graphical or algebraic method. Here let’s use the algebraic method.

The algebraic method is of three types:
1. Substitution Method
2. Elimination Method
3. Cross Multiplication Method
So by using any of the above methods we can solve the given pair of linear equations.

Complete step by step solution:
Given

x−2y=−6.............(i) \-x+y=2.............(ii) x - 2y = - 6.............\left( i \right) \\\ \- x + y = 2.............\left( {ii} \right) \\\

Now since we are using an algebraic method let’s use the substitution method given above to solve the question.

Substitution Method:
In this method from the given two equations of two variables, we have to substitute the equation of any one variable from one of the equations and then substitute it in the other one such that the second equation becomes an equation of one variable and thereby we can solve for that one variable.

So by using the above definition, let’s substitute x from equation (i) to (ii):
⇒x−2y=−6 ⇒x=−6+2y..................(iii)  \Rightarrow x - 2y = - 6 \\\ \Rightarrow x = - 6 + 2y..................\left( {iii} \right) \\\

Now substituting (iii) in (ii):
⇒−x+y=2\Rightarrow - x + y = 2

Now substituting (iii), we get:

⇒−(−6+2y)+y=2 ⇒6−2y+y=2 ⇒6−y=2..................(iv)  \Rightarrow - \left( { - 6 + 2y} \right) + y = 2 \\\ \Rightarrow 6 - 2y + y = 2 \\\ \Rightarrow 6 - y = 2..................\left( {iv} \right) \\\
On observing (iv) we get that it’s an equation of only ‘y’ such that we can solve for the variable ‘y’.

⇒−y=2−6 ⇒−y=−4 ⇒y=4......................(v)  \Rightarrow - y = 2 - 6 \\\ \Rightarrow - y = - 4 \\\ \Rightarrow y = 4......................\left( v \right) \\\
So now we get:y=4y = 4

Now substituting (v) in (iii) to get the value of ‘x’:
⇒x=−6+2y ⇒x=−6+2(4) ⇒x=−6+8 ⇒x=2.........................(vi)  \Rightarrow x = - 6 + 2y \\\ \Rightarrow x = - 6 + 2\left( 4 \right) \\\ \Rightarrow x = - 6 + 8 \\\ \Rightarrow x = 2.........................\left( {vi} \right) \\\

Therefore on solving x−2y=−6    and    −x+y=2x - 2y = - 6\;\;{\text{and}}\;\; - x + y = 2 algebraically we get x=2    and    y = 4.x = 2\;\;{\text{and}}\;\;{\text{y = 4}}{\text{.}}

Note: While solving a pair of linear equations one should take care of following things:

We need to express the two linear equations in two different variables, we can solve them either by substitution, elimination, cross multiplication method or by graphical method.

We can also check the validation of the ‘x’ and ‘y’ values by putting them in the given equations and checking whether it satisfies mathematically or not.