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How do you solve $x - 2y = 5$ and $2x - 4y = 1$ using substitution ?

Last updated date: 23rd Jul 2024
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Hint: In order to solve this question , we need to have knowledge about the concept of Substitution method and with the help of this we will solve it step by step . Substitution method comes under one of the categories of algebraic methods to solve simultaneous linear equations . What we actually do in this method is substitution, that is the value of one variable from one equation is substituted in the other equation . Also we will perform some calculations to simplify the given equation by somewhere using equivalent equations . Equivalent equations are said to be algebraic equations that may have the same solutions if we add or subtract the same number to both sides of an equation - Left hand side or Right hand side of the equal sign . Or we can multiply or divide the same number to both sides of an equation - Left hand side or Right hand side of the equal to sign with the method of simplification .

To solve the given question by using substitution , we will first choose any one equation and solve it for one of its variables .
Like here , we chose the first equation $x - 2y = 5$ and we will solve for any one variable from the two variables $x$ or $y$ . Here , we would like to solve for $x$ , then we get –
$x - 2y = 5 \\ x = 2y + 5 \;$
Now, we will substitute the value of $x$ we just solved for in the second equation so that there will be only one variable in the equation and it will be easy for us to solve the equation with one variable.
$2x - 4y = 1$
Substituting $x = 2y + 5$ in the above equation we get ,
$\Rightarrow 2x - 4y = 1 \\ \Rightarrow 2(2y + 5) - 4y = 1 \\ \Rightarrow 4y + 10 - 4y = 1 \;$
You can perform calculations only between like terms . In like terms The Coefficients can be different but should have the same variable with the same power .
So, -4y gets cancelled .
$2x - 4y = 1 \\ 2(2y + 5) - 4y = 1 \\ 4y + 10 - 4y = 1 \\ 10 = 1 \;$
Which is Not True .
Hence , no solutions for the given equations .
So, the correct answer is “Option B”.

Note: In equivalent equations which have identical solutions we can perform multiplication or division by the same non-zero number both L.H.S. and R.H.S. of an equation. In like terms The Coefficients can be different but should have the same variable with the same power . Always remember you can perform calculations only between like terms .Cross check the answer and always keep the final answer simplified .
Remember the algebraic methods and apply appropriately .